Global search should clear badges as soon as the related search term is deleted or changed

### What Happened?

After performing a global search with matches, the search term in the header bar was deleted. The badges remained but there is no longer any information as to what search they refer.

### Steps to Reproduce

- Perform a global search with matches
- Delete the search term
- Observe the badges remain

### Expected Behavior

The badges should disappear as soon as the search term to which they relate is changed.

### OS Version

8.x (Circe)

### OS Architecture

amd64 (on most hardwares)

### Session Type

Secure Session (Wayland)

### Software Version

Latest release (I have run all updates)

### Log Output

```shell

```

### Hardware Info

_No response_