The search bar text gets cleared when switching between package types in the Package Manager window

Reproduction steps:
1. Create any project
2. In the Menu bar select Window → Package Manager
3. In the Package Manager window enter any text in the search bar
4. Change the package type (e.g. from "In Project" to "Unity Registry")
5. Observe the search bar

Expected result: The text entered in the search bar should persist across different package types.
Actual result: The text entered in the search bar is cleared when switching between package types.

Reproducible with: 2021.3.44f1, 2022.3.48f1, 6000.0.21f1

Reproducible on: Windows 11
Not reproducible on: No other environments tested

Note: In Unity 6000.0.21f1 the text is only saved in the Package Type search bar where it was written, as the issue is that the search prompt is always lost when switching through the types