`After /purge_history users still get pushed for new messages` flake

Nobody has claimed this yet.

https://github.com/matrix-org/synapse/actions/runs/6070182860/job/16465779518

```
# Started: 2023-09-04 07:17:01.879
# Ended: 2023-09-04 07:17:04.922
# Got JSON::number(5), expected 2 for at tests/48admin.pl line 576.
# 0.054136: Registered new user (via secret) @anon-20230904_070440-729:localhost:8839
# 0.120044: Registered new user @anon-20230904_070440-730:localhost:8839
# 0.843307: User @anon-20230904_070440-729:localhost:8839 joined room
# { room_id => "!qzxCvZZbDkHNBlGJSU:localhost:8839" }
# 1.123424: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $ny1MQUwhG0FjvOl6jznwUAJJDK-NFY1vOO-_90l0KUU
# 1.206519: Receipt
# {
# content => {
# "\$ny1MQUwhG0FjvOl6jznwUAJJDK-NFY1vOO-_90l0KUU" => {
# "m.read" => {
# "\@anon-20230904_070440-729:localhost:8839" => { ts => JSON::number(1693811823049) },
# },
# },
# },
# type => "m.receipt",
# }
# 1.308130: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $r1D9MwKGJX99KvYlAgSaKVgdv7ZfX15te-pvKoEYJsc
# 1.440464: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $n2bno8AVIAOQ6Q6o1uxyRNrRFYty199kAGnQVxcqA2E
# 1.564649: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $6g5ts8hBj13k2sWAV1e11opf6lFMuYRAjkYKI-3j3H0
# 1.702517: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $cnGoq3nweDJU30uWbYnFHwn6KnJFD2LFWGk2bNjHJzs
# 1.821504: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $46a07kzdvTy0Lb7u-GkeZrAAKtgPjQ_5c-uap-b_TVo
# 2.115693: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $cSS0SI3_hA-DcLpERaQjFHN1ndkgmrmEQIUjBa_J5BI
# 2.233397: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $IKQSoLOWCBnNjH-uN1p5yiAP0I2K0RdGWTEgURLTMZg
# 2.366796: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $kWE-Gb-_ilDtcT16SNdX1H6-gwg_3HWu4aYwAJT0b4I
# 2.483043: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $rLXInmilT8rCxtlFVSaWLBmw624LF1hNVEL7L6kxf_8
# 2.596196: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $Esvx7Vh65jkejQsnyMNgKKfyQVKRI2683bFtTBQGQPU
# 2.647668: Sync for @anon-20230904_070440-730:localhost:8839
# {
# "device_one_time_keys_count" => { signed_curve25519 => JSON::number(0) },
# "device_unused_fallback_key_types" => [],
# "next_batch" => "s4412_861_14_11_52_3_18_780_0_5",
# "org.matrix.msc2732.device_unused_fallback_key_types" => [],
# "presence" => {
# events => [
# {
# content => {
# currently_active => JSON::true,
# last_active_ago => JSON::number(844),
# presence => "online",
# },
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.presence",
# },
# {
# content => {
# currently_active => JSON::true,
# last_active_ago => JSON::number(1474),
# presence => "online",
# },
# sender => "\@anon-20230904_070440-729:localhost:8839",
# type => "m.presence",
# },
# ],
# },
# "rooms" => {
# join => {
# "!qzxCvZZbDkHNBlGJSU:localhost:8839" => {
# account_data => { events => [] },
# ephemeral => {
# events => [
# {
# content => {
# "\$ny1MQUwhG0FjvOl6jznwUAJJDK-NFY1vOO-_90l0KUU" => {
# "m.read" => {
# "\@anon-20230904_070440-729:localhost:8839" => { ts => JSON::number(1693811823049) },
# },
# },
# },
# type => "m.receipt",
# },
# ],
# },
# state => { events => [] },
# summary => {},
# timeline => {
# events => [
# {
# content => { body => "Message 1", msgtype => "m.text" },
# event_id => "\$r1D9MwKGJX99KvYlAgSaKVgdv7ZfX15te-pvKoEYJsc",
# origin_server_ts => JSON::number(1693811823114),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(1409), transaction_id => 976 },
# },
# {
# content => { body => "Message 2", msgtype => "m.text" },
# event_id => "\$n2bno8AVIAOQ6Q6o1uxyRNrRFYty199kAGnQVxcqA2E",
# origin_server_ts => JSON::number(1693811823235),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(1288), transaction_id => 977 },
# },
# {
# content => { body => "Message 3", msgtype => "m.text" },
# event_id => "\$6g5ts8hBj13k2sWAV1e11opf6lFMuYRAjkYKI-3j3H0",
# origin_server_ts => JSON::number(1693811823369),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(1154), transaction_id => 978 },
# },
# {
# content => { body => "Message 4", msgtype => "m.text" },
# event_id => "\$cnGoq3nweDJU30uWbYnFHwn6KnJFD2LFWGk2bNjHJzs",
# origin_server_ts => JSON::number(1693811823496),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(1027), transaction_id => 979 },
# },
# {
# content => { body => "Message 5", msgtype => "m.text" },
# event_id => "\$46a07kzdvTy0Lb7u-GkeZrAAKtgPjQ_5c-uap-b_TVo",
# origin_server_ts => JSON::number(1693811823631),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(892), transaction_id => 980 },
# },
# {
# content => { body => "Message 6", msgtype => "m.text" },
# event_id => "\$cSS0SI3_hA-DcLpERaQjFHN1ndkgmrmEQIUjBa_J5BI",
# origin_server_ts => JSON::number(1693811823920),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(603), transaction_id => 981 },
# },
# {
# content => { body => "Message 7", msgtype => "m.text" },
# event_id => "\$IKQSoLOWCBnNjH-uN1p5yiAP0I2K0RdGWTEgURLTMZg",
# origin_server_ts => JSON::number(1693811824048),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(475), transaction_id => 982 },
# },
# {
# content => { body => "Message 8", msgtype => "m.text" },
# event_id => "\$kWE-Gb-_ilDtcT16SNdX1H6-gwg_3HWu4aYwAJT0b4I",
# origin_server_ts => JSON::number(1693811824169),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(354), transaction_id => 983 },
# },
# {
# content => { body => "Message 9", msgtype => "m.text" },
# event_id => "\$rLXInmilT8rCxtlFVSaWLBmw624LF1hNVEL7L6kxf_8",
# origin_server_ts => JSON::number(1693811824293),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(230), transaction_id => 984 },
# },
# {
# content => { body => "Message 10", msgtype => "m.text" },
# event_id => "\$Esvx7Vh65jkejQsnyMNgKKfyQVKRI2683bFtTBQGQPU",
# origin_server_ts => JSON::number(1693811824410),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(113), transaction_id => 985 },
# },
# ],
# limited => JSON::true,
# prev_batch => "s4402_861_14_11_52_3_18_780_0_5",
# },
# unread_notifications => {
# highlight_count => JSON::number(0),
# notification_count => JSON::number(0),
# },
# },
# },
# },
# }
# 2.650694: Found event $Esvx7Vh65jkejQsnyMNgKKfyQVKRI2683bFtTBQGQPU for @anon-20230904_070440-730:localhost:8839
# 2.650715: Got message down sync
# 3.000466: Sent 'm.room.message' event in !qzxCvZZbDkHNBlGJSU:localhost:8839: $p3o6N3VdmT0If1wWbXvv2XJiaGMYUok3KdwKGU15Rv8
# 3.040547: Room
# {
# account_data => { events => [] },
# ephemeral => { events => [] },
# state => { events => [] },
# summary => {},
# timeline => {
# events => [
# {
# content => { body => "New Message", msgtype => "m.text" },
# event_id => "\$p3o6N3VdmT0If1wWbXvv2XJiaGMYUok3KdwKGU15Rv8",
# origin_server_ts => JSON::number(1693811824802),
# sender => "\@anon-20230904_070440-730:localhost:8839",
# type => "m.room.message",
# unsigned => { age => JSON::number(115) },
# },
# ],
# limited => JSON::false,
# prev_batch => "s4412_861_14_11_52_3_18_780_0_5",
# },
# unread_notifications => {
# highlight_count => JSON::number(0),
# notification_count => JSON::number(5),
# },
# }
Totals: 813 passed, 11 expected fail, 1 failed
```

Start with tests/48admin.pl around line 576 and the linked GitHub Actions run to reproduce the failure. Compare the expected notification count of 2 with the observed count of 5 after the history purge, and consider the test complete when this scenario no longer flakes and the assertion reliably matches the intended behavior.

Written by the indexing model from the issue text.