Focused Inspectors Window Label Icon is not updated when the Icon for the Game Object is changed unless the Inspector is reopened

Steps to reproduce:

1. Open any Project
2. Select any Game Object in the Hierarchy
3. In the Inspector, open a focused Inspector of the same Game Object from the Kebab menu at the top right corner by choosing "Properties..."
4. Click on the cube framework icon to the left of the Game Objects Label and choose any color icon
5. Observe the Focused Inspector tab Label, close the Focused Inspector, open it again, and observe the label again

Actual results: Focused Inspectors Window tab Label is only updated when it is reopened

Expected results: Label icon is updated immediately

Reproducible with versions: 2021.3.43f1, 2022.3.45f1, 6000.0.17f1

Tested on (OS): macOS Sonoma, Windows 11