Search query asset is not on the “Project” list in the Search window if it was duplicated

How to reproduce:
1. Open the user’s attached “bug-asset-search.zip” project
2. Open the Search window
3. In the search field, insert “testing” text
4. Press the save button and choose “Save Project”
5. In the Project window, select “testing.asset” and duplicate it
6. Observe the “Project” tab in the Search window

Expected result: “testing 1” is on the list
Actual result: “testing 1” is not on the list

Reproducible with: 2021.3.28f1, 2022.3.4f1, 2023.1.2f1, 2023.2.0a21

Reproduced on: macOS 13.2.1 (Intel)

Notes:
After restarting the editor then, the duplicated query asset is visible in the “Project” tab
Also reproducible if duplicating query asset in the file explorer