Browser Extension shows locked even if unlocked

### Steps To Reproduce

1. Have at least 2 browser windows open on different monitors.
2. Log in to the browser extension in one window.
3. In the window where you logged in, it appears unlocked. The other window still has a lock icon.
4. If you now click on the window that still has the lock on it, it will immediately appear unlocked.

### Expected Result

Even if I have browser windows open on other monitors, the icon should update everywhere even if the windows are not in focus.

### Actual Result

The Bitwarden icon continues to show the lock on windows that are not in focus.

### Screenshots or Videos

_No response_

### Additional Context

_No response_

### Operating System

Windows

### Operating System Version

10 x64 (19045.5854)

### Web Browser

Vivaldi

### Browser Version

7.4.3684.38 x64

### Environment Versions

2025.5.0

### Issue Tracking Info

- [ ] I understand that work is tracked outside of Github. A PR will be linked to this issue should one be opened to address it, but Bitwarden doesn't use fields like "assigned", "milestone", or "project" to track progress.

Reproduce the issue in the Bitwarden browser extension with two windows on different monitors, using the Windows and Vivaldi versions listed. Trace the lock icon's update behavior when the window is unfocused; done means all open windows update immediately after login or unlock without needing focus.

Written by the indexing model from the issue text.