Changes to display names do not apply consistently to all rooms

### Description

When changing display name the change applies to certain rooms, but not all the rooms I'm in.

### Steps to reproduce

- change display name via Element iOS app or Element desktop app (same issue happens regardless of client)
- change applies without errors but not consistently to every room

### Homeserver

element.io

### Synapse Version

n/a

### Installation Method

Docker (matrixdotorg/synapse)

### Database

n/a

### Workers

I don't know

### Platform

n/a

### Configuration

_No response_

### Relevant log output

```shell
Do not have access to logs
```

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

_No response_