This afternoon as a result of posting on the Apple Pro XDR Display Forum the problem of my Apple Pro XDR display not updating to the new firmware (2.2.2) an Apple Support Agent reached out to me by phone. She carefully took down everything that I had done, collected a variety of system reports including one specific to the Pro XDR display, and promised to escalate the case to a Senior Support Specialist. She also ask me to click on the Apple>About this Mac>Software Update>Advanced and check Install System Data Files and Security Updates (which had not been checked).
Tonight I sat down to do some image editing in Lightroom. I woke my Mac Pro from sleep and tried to connect to my QNAP NAS server where all of my image files are saved. I found that I could not connect to the NAS. This has happened occasionally in the past and I have always solved it by restarting the Mac Pro and then connecting to the NAS. However, before I could restart the Mac Pro I had a fleeting glimpse of a message in the top right of the Pro XDR display that disappeared before I could read it. When I restarted the Mac Pro the display remained dark and did not turn on. After over 1 minute of waiting, I did a power button forced reboot of the Mac Pro. The Mac Pro booted and the display was very slow (ca. 1 minute) to turn on, but when it did appear all seemed normal. I went to Apple > About This Mac>System Report>Graphics/Display and saw that my display was now on firmware version 2.2.2 so somehow during all of this the firmware update had finally been accomplished. And, yes, I could readily connect to my NAS after this reboot.
I would suggest checking Install System Data Files and Security Updates as described above to anyone having a problem getting their Pro XDR display to update its firmware.