I understand that the max speed is 1333mhz in a Mac Pro even if the RAM is capable of faster speeds. My Mac Pro specs are in my sig. I have a dual X5680 4,1 => 5,1. I don't think the 4,1 with updated firmware vs. a genuine 5,1 is particularly relevant as I am able to run 16gb modules at 1333mhz no problem.
Looking back at previous posts it appears that the the secret to getting 256gb of RAM to run at 1333mhz in macOS is to specify the RAM config with open core using Clover. Too bad I sold my 32gb modules after I was unable to get them to work in macOS.
I wonder if the 256gb@1333mhz config is stable when you actually ramp up the memory pressure. It's one thing to trick the About dialog to say 1333mhz, and another thing to trick the Mac to actually perform (without crashing) at 1333mhz. In Windows 10 Pro I was only able to get the 32gb RAM modules to show 1066mhz which as I understand is a limitation of using quad-rank memory. The Linux screen shots you posted do not show a memory speed so I'm curious what speed the RAM runs at using Linux. AFAIK there's only one type of 32gb RAM module you can actually buy: quad rank rdimm. This means that all 32gb memory modules ought to be limited to at most 1066mhz, or 800mhz with a full 256gb, but I would be happy to be wrong about that.
Either way your Danish contact seems to have achieved something that was previously considered impossible, booting macOS with 32gb RAM modules installed which is nice to see. I never tested on Big Sur as it was not out yet at the time but I did test on Catalina and macOS would not boot with even a single 32gb RAM stick installed. From the screenshots you posted earlier it seems like your contact was able to get Catalina to boot as well so I'm guessing the same open core clover config is able to work using Catalina as well.