Spotlight search does not populate based on later sync in background

### Steps to reproduce

1. Accept an invite on Element X, have a nice chat, etc.
2. Open Element Desktop
3. Hit Ctrl+K as soon as it's not completely blank
4. Start typing a search for the new room from step on
5. Wait for sync to finish

### Outcome

#### What did you expect?

The search result is found at least after sync has finished catching up in background

#### What happened instead?

Spotlight index does not update

### Operating system

Arch

### Application version

Element Nightly version: 2025122201 Crypto version: Rust SDK 0.16.0 (8471012), Vodozemac 0.9.0

### How did you install the app?

aur

### Homeserver

synapse 1.143.0

### Will you send logs?

No

Start by reproducing the Ctrl+K search flow while background sync catches up, then trace where the Spotlight index is updated after sync. Done means a newly synced room becomes searchable without reopening Spotlight or the application.

Written by the indexing model from the issue text.