Apple Studio Display stays black when is connected to one of MBPs

Hello. There are:

• 2 Macbook Pro 16" 2019 (initially Monterey 12.6.3, during troubleshot both are upgraded to Sonoma 14.4)
• a new Apple Studio Display

After unpacking, the display was connected to one of MBPs and it still works just fine with it. But when I'm trying to connect it to another MBP, the laptop starts charging from the display and "System Report" detects the display as connected but the display remains black and is not shown in the "Displays" settings. The display works fine if the system is booted to Safe Mode.

What I've tried so far (without success):

• Upgrade to Sonoma
• Update the display firmware to latest version
• Delete the "windowserver" configuration files as per Apple Studio Display black after selectin… - Apple Community
• Reboot the display, replace the TB cable, use different USB-C ports on both the MBP and the display
• Reinstall the OS using Recovery Mode, i.e. not from scratch

It seems that something is broken in the macOS installation. Reinstalling it from scratch would likely help but is not really an option for me.

Any ideas how to fix it?