Today I had the time to try few things.
First I "Airplayed" videos from the Mini to the Apple TV 4K which is on the upper floor. The connection worked, however I am not sure about the quality. This is a screenshot that shows that the maximum resolution I could get was 1080p. I did not have a lot of time so I did not try to verify if I could change any of the available settings. Changing to HDR however did not make any difference.

Then I tried connecting the Mini to the LG Monitor with the same HDMI cable that I am using with the Apple TV. It is a 6' Belkin that I purchased with the Apple TV from Apple. It is not 3" as recommended but that's all I had. Using the HDMI instead of Display Port did not make any difference.
At that point I decided to try the LG monitor with the Apple TV. It worked perfectly right away. All whites were whites. No dull colors. I tried the same movie and the HDR seemed to work perfectly. That would indicate that the LG monitor is not the problem.
Just to make sure, I finally connected the Mac mini to the Samsung TV, the one that I use with the Apple TV. Unfortunately again, with he HDR setting it showed the exact same problems.
Again, I did not use a 3’ HDMI cable, however the USB-C to Display Port cable that I normally use is 3’. I could buy a 3’ HDMI cable just to make sure, but I tend to exclude that the cable is the problem. All seems to indicate that the problem is with the Mac Mini. As hcsitas said it may be due to a hardware limitation, but want I to hope that is just a matter of setting or software.
I am wondering if members of the forum have been able to run the HDR successfully with their Mini’s.