Steps to reproduce:

1. Open new project
2. Import [https://assetstore.unity.com/packages/3d/characters/humanoids/lowpoly-survival-character-rio-273074] (any other asset with an avatar would work. Avatars allow the easiest reproduction of bug)
3. Open Search Window by pressing the magnifying glass at top right of editor or by pressing the shortcut ctrl + k
4. Ensure 'Toggle Query Builder Mode' is active. If not, press the toggle button or use the shortcut F1
5. Press the + button in the Search Field and type in Animator
6. Click the green 'Avatar (Avatar)' option in the menu to add the animator.avatar pill to the search query
7. Open the object picker on the pill. A Select Avatar picker will appear. Select an avatar (RioAvatar if using specified asset store asset from above)
8. Toggle Query Builder Mode off and then back on again
9. Re-open the object picker on the pill

Actual results: The object picker has turned from a 'Select Avatar' picker into a generic 'Select GameObject' picker

Expected results: The object picker remains as a 'select Avatar' picker

Reproducible with versions: 

• 6000.0.10f1
• 2023.2.20f1
• 2022.3.36f1

Not reproducible with versions: -

Can’t test with versions: 

• 2021.3.40f1 (Doesn't have Query Builder functionality)

Tested on (OS): Windows 11

Notes:

• A particular pain point is that the bug occurs with Saved Searches. If a search with an object picker (like Avator) is saved, then when the search is re-loaded it will have the generic 'Select GameObject' picker.
• Does not happen with all object picker types.