AirPods Pro 3 case firmware not updating, causing issues

I’ve been having quite a lot of issues with my AirPods Pro 3. I recently noticed that a new firmware update had been released, and the earbuds updated successfully. However, I then realized that the charging case is still running a very old version. The earbuds are now on version 8B40, but the case is still on 8B34, so it seems that several updates have not been applied to the case.

Does anyone happen to know a trick to get the case updated to the same version as the earbuds? I suspect that many of the issues I’ve experienced might actually be caused by this.

For example, it feels like the case is constantly trying to pair with nearby devices. Even when the earbuds are inside the case, I almost always get the AirPods popup on my phone whenever the screen wakes up.

On top of that, the battery drains very quickly, even when the earbuds are not in use and are just sitting in the case.

Another recent issue is that if both earbuds are not in use, the connection to my phone becomes unstable if the case and one earbud are farther away. At home, I often use just one earbud so I can still hear if my family needs something, but this setup has started causing connection problems.