room alias lookups on `libera.chat` timing out

This issue has been migrated from [#14462](https://github.com/matrix-org/synapse/issues/14462).

---

### Description

I created a room yesterday, and no client can display it because getting `client/v3/directory/room/` times out or crashes with a 502 error.

With curl:

```
$ curl https://matrix.org/_matrix/client/v3/directory/room/%23valtest8:libera.chat
{"errcode":"M_UNKNOWN","error":"Failed to fetch alias"}
```

With my own client, the connection times out

And from my browser (regardless of being logged in), I'm getting Cloudflare's 502 Bad Gateway page

### Steps to reproduce

- join a room on a different server, with no other user from your own homeserver in the room
- leave and re-join a couple of times
- Eventually the client will be stuck

### Homeserver

matrix.org

### Synapse Version

1.71.0rc2 (b=matrix-org-hotfixes,39ea774cff,dirty)

### Installation Method

_No response_

### Platform

n/a

### Relevant log output

```shell
n/a
```

### Anything else that would be useful to know?

_No response_

Start at the client/v3/directory/room/ endpoint and reproduce the room-alias lookup using a room on a different server, following the join, leave, and re-join sequence described. Investigate why the lookup times out, crashes, or returns a 502, and consider the issue done when the alias lookup completes successfully without those failures.

Written by the indexing model from the issue text.