NATS timeout when sending data

*This issue was originally created by [**@kegsay**](https://github.com/kegsay) at .*

```
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.119204493Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=jUzhbATP6X9S req.method=PUT req.path="/_matrix/client/r0/user/@rek2:hispagatos.org/account_data/im.vector.setting.breadcrumbs" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.120056708Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=i2RDM6p0K8WL req.method=POST req.path="/_matrix/client/r0/rooms/!0bHdZkTyCe4eNfIH:hispagatos.org/read_markers" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.121249160Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=H51vUbR5ekqW req.method=PUT req.path="/_matrix/client/r0/user/@rek2:hispagatos.org/account_data/im.vector.setting.breadcrumbs" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.122014486Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=PFUrDPcD51VI req.method=POST req.path="/_matrix/client/r0/rooms/!fsheNkKUDhptJdEuhX:matrix.org/read_markers" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.123378900Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=3k4tGBLYmw74 req.method=PUT req.path="/_matrix/client/r0/user/@rek2:hispagatos.org/account_data/im.vector.setting.breadcrumbs" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.124253392Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=Y96R3b7T4Aqe req.method=POST req.path="/_matrix/client/r0/rooms/!Omo8TLnJpogmnwnZ:hispagatos.org/read_markers" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.125588658Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=7qqEJzBulQgo req.method=PUT req.path="/_matrix/client/r0/user/@rek2:hispagatos.org/account_data/im.vector.setting.breadcrumbs" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.126389076Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=DWk43Dv5Vqou req.method=PUT req.path="/_matrix/client/r0/user/@rek2:hispagatos.org/account_data/im.vector.setting.breadcrumbs" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.127673324Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=SQPKZRuWWhlg req.method=POST req.path="/_matrix/client/r0/rooms/!NhYYJsDjWnGSihypYQ:matrix.org/read_markers" user_id="@rek2:hispagatos.org"
Jan 15 17:58:49 matrix dendrite-monolith-server[23116]: time="2022-01-15T17:58:49.128379151Z" level=error msg="syncProducer.SendData failed" error="nats: timeout" req.id=Om1zxNhwJxUr req.method=POST req.path="/_matrix/client/r0/rooms/!0bHdZkTyCe4eNfIH:hispagatos.org/read_markers" user_id="@rek2:hispagatos.org"
```

The report names syncProducer.SendData and shows repeated nats: timeout errors from the monolith server; start by tracing that entry point and the NATS messaging path. Reproduce the timeout, identify its cause, and add regression coverage before considering the issue done.

Written by the indexing model from the issue text.