Safari "private" window are not moved by action "Move Window To Desktop x"

I have configured shortcuts for action "Move Window to Desktop x" (x in 1, 2, 3, etc.)
It works well except for Private Safari windows. Normal Safari windows, as well as all other applications I've tried, are moved correctly.

Same for actions "Move window One Space/Desktop Right" (and Left): all windows are moved, except Safari Private ones.

Affects 2 Mac systems (Mac Mini and MacBookPro), both running latest Catalina (10.15.6).
BTT version 3.400

How to reproduce:

  • create shortcut for moving window to other desktop
  • open Safari, open new private window (shift-command-N)
  • press the shortcut keys that are supposed to move the window