I’ve installed the update to Photo AI 3.2.1 without any issues during the installation process.
After the first tests I’ll see no problems.
The plugin in LrC is working fine (Lightroom 13.5.1, Windows 11 23H2).
Also the differences of brightness, saturation and the little color shift between original and the image from TPAI as decribed here Topaz Photo AI v3.2.0 - #13 by axel.matthies are gone.
As far as I can see, this is a version which I can work with.