The chip that it uses for driving the displays is the DisplayLink DL-6950, which is the same chip as what’s in the Pluggable UD-6950PDZ that you linked.
Here’s a suggestion for something to try:
In the AMD drivers under display create a new custom resolution. Fill in 4k 60 Hz and select “CVT - Reduced Blanking” under timing standard. Do this for both displays.
That will create a separate resolution setting that has less blanking than normal (blanking is bandwidth that is deliberately wasted in order to improve compatibility with old displays). Mathematically dual 4k 60 Hz with the normal amount of blanking is just a bit beyond what the DisplayPort link is capable of, so conserving bandwidth by reducing blanking may be able to get it working.
Edit: Sorry, I missed that you’re on Linux. I don’t know how to do my suggestion on Linux, only on Windows.