Display names update but then revert on refresh

### Steps to reproduce

1. Have two users that share a room
2. Change display name of user 1. See that it updates on user 2's screen
3. Refresh user 2 client shortly after.

### Outcome

#### What did you expect?
User 2 sees user 1's new name before & after refresh

#### What happened instead?
After refresh, user 1's display name reverts to whatever it was before it was changed. Clear cache & reload fixes it.

### Operating system

_No response_

### Browser information

_No response_

### URL for webapp

_No response_

### Application version

_No response_

### Homeserver

_No response_

### Will you send logs?

Yes

Reproduce the two-user room scenario in the web client, then trace how a changed display name is persisted and restored during refresh. Done means user 2 continues to see user 1's new name after refreshing without clearing the cache.

Written by the indexing model from the issue text.