Hi,
You try to look at the referenced registry key (\SOFTWARE\Microsoft\Windows NT\CurrentVersion\ProductName) and you see that Windows 11 thinks it is Windows 10.
If so, then it's not a BGinfo problem, it's a windows11 problem
This should be the reason why Microsoft has not updated it. It is recommended to find a suitable replacement, or wait for the next version of Windows 11 to test again, and the next version will be released in the second half of 2022.