AppCenter notification dot in the Dock doesn't go away after successful update

### What Happened?

When there are available updates in AppCenter and you have successfully installed them all, the notification dot doesn't go away automatically. You have to close and open AppCenter again manually for it to run a check for updates again and only then does the notification dot go away.

Before:


After:


### Steps to Reproduce

1. AppCenter indicates it has updates available with the notification dot in the Dock
2. Click to open AppCenter which shows the available updates
3. Install the updates
4. Notice that the notification dot in the Dock on the AppCenter icon doesn't go away

### Expected Behavior

Upon a successful completion of the updates the notification dot should no longer be shown.

### OS Version

8.x (Circe)

### OS Architecture

amd64 (on most hardwares)

### Session Type

Secure Session (Wayland, This is the default)

### Software Version

Latest release (I have run all updates)

### Log Output

```shell

```

### Hardware Info

_No response_

Start by reproducing the issue in AppCenter on elementary OS 8.x: install all listed updates and observe whether the Dock notification dot remains. Trace the update-completion flow and its update check, then verify that the dot disappears immediately after successful installation without reopening AppCenter.

Written by the indexing model from the issue text.