First, I wanted to confirm as, after having activate the MixConsole right zone, indeed, the tabs content were swapped, but the tabs display wasn’t following. The two shortcuts defined being :
- Ctrl + Alt + ► (Show next tab)
- Ctrl + Alt + ◄ (Show previous tab)
Then , for a reason, I activated the ‘center’ MixConsole area and went back to highlight the right zone again : the result is what we should expect, then : both tabs display and their content were synchronized.
So, at the end, I don’t know what to think of this one : indeed, it seems that there is an issue, as on my end, I actually had to highlight twice the right zone to make it work as expected… 
FWIW : tested on C13.0.10, Windows 10 Pro 22H2.