Mark as unread is not reliable persistent, does not trigger a Badge Icon on the App Icon and chats don't get pinned on top of the chat list

### Steps to reproduce

1. Mark a chat as unread (new feature in version 1.10.2)

### Outcome

#### What did you expect?

1. I did expect that the chat get pinned at the top of the chat list (since I have turned on the option "Pin rooms with unread messages")
2. I did expect that the app icon on my home screen shows a badge icon for a unread message
3. I did expect that the unread state is persistent between app relaunches

#### What happened instead?

1. The chat stays in the chat where it was
2. The app icon does not show a badge icon
3. The unread state is lost after app relaunch

### Your phone model

iPhone XR

### Operating system version

16.3

### Application version

1.10.2

### Homeserver

_No response_

### Will you send logs?

No

Reproduce the mark-as-unread flow on an iPhone XR running iOS 16.3 with app version 1.10.2, then trace the handling of unread state, room ordering, and app-icon badges. Done means the marked chat stays unread after relaunch, moves to the top when the pinning option is enabled, and produces an app-icon badge.

Written by the indexing model from the issue text.