I ordered and customised this compression ring on Aliexpress. The headset is now 100% free of play (without any torque struggles, just as it's meant to be and ever was on any bike). It seems to be the Specialized design. I hope that it is also a bit gentler for the steerer tube.
referring to the video by henrychungfitness:
https://www.aliexpress.us/item/3256805803796644.html

I had to widen the two openings a little so that the Shimano brake cables would fit through.

I cut off the outer plastic ring on the headset cap so that it doesn't rub on the frame.