Site Selector Drop-Down does not show current site value correctly

[peternazarov]

Describe the bug
Site Selector Drop-Down does not show current site value correctly. It is shown correctly only when a page is loaded for the first time after the application pool process is started. Then when you switch to another site the current site (the highlighted site name in the Site Selector Drop-Down) does not change.

Sitecore Version
9.1

To Reproduce
Steps to reproduce the behavior:

1. Restart App Pool
2. Load homepage
3. Suing Site Selector switch to another site
4. Open Site Selector and observe the current (selected/highlighted) site name - it does not match with the current site.

Expected behavior
When you click on Site Selector the drop-down list shows the current site name highlighted.

Screenshots