Hi! I understand your PBS app on your ChromeOS device is asking you to update the Play Store app, even after you've already updated it. Let’s get this sorted out.

A couple of clarifying questions first:
- When exactly does this message appear? Is it immediately when you try to open the PBS app or after some loading?
- Have you tried closing and reopening both the PBS app and the Play Store app?
- Also, have you restarted your ChromeOS device since updating the Play Store?

Meanwhile, here’s a step-by-step you can try based on your situation:

1. Close and Reopen Apps: Fully close the PBS app and Google Play Store app from your device and then try reopening PBS again. Sometimes the apps need a fresh start.

2. Restart Your ChromeOS Device: A restart helps apply updates properly and resets app states. This is often a key step for update-related errors.

3. Clear Cache for Play Store:
- Go to your ChromeOS Settings.
- Find Apps, then Manage Apps, and locate Google Play Store.
- Clear the cache (and if needed, clear data as a last resort).
This can remove any corrupted data causing the app to think it’s outdated.

4. Check for System Updates: Ensure your ChromeOS itself is fully updated. Sometimes Play Store issues stem from OS-version incompatibilities.

5. Reinstall the PBS app: If none of the above works, uninstall and reinstall the PBS app. This refreshes its dependencies on system apps like Play Store.

If after these steps you still get the same message, there may be a deeper conflict between the Play Store version and the PBS app on your device. Let me know what you’ve tried already or if any step is unclear, and I’ll help you deeper.

Are you comfortable trying these steps?