[MaterialDatePicker] Selected month is lost when app returns from background

[mtotschnig]

Description: Issue can be reproduced with Catalog app, as seen in attached recording

1. Open DatePicker, and navigate to month in the past or in the future.
2. Put app into background and then bring to foreground
3. DatePicker selection moves back to current month (or the one it was opened with initially)

Expected behavior: State of the component should be maintained.

The problem only happens in background/foreground use case. When device is rotated, the state is correctly maintained.

Android API version: Observed on multiple versions, e.g. 30 and 35.

Material Library version: Catalog app built from HEAD on master branch (690bbfc). Also observed with 1.12.0

Device: Observed on multiple devices, e.g. Android emulator, Pixel 6a

As a user of my app has reported (mtotschnig/MyExpenses#1626), this has important UX implication: User navigates to a given month, and before selecting the day, they need to verify information in a different app, and when returning to my app, the month they already have selected, is lost.

Screen_recording_20250116_235244.mp4

[mtotschnig]

On a technical level, the problem seems to be caused by the fragments being recreated every time in onStart. Reusing the existing fragments seems to solve the issue:
mtotschnig@48d4f39

[mtotschnig]

If someone from the team confirms that my solution is valid, I can open PR.