Hello, for the record, I am using this kind of controller, which seems pretty generic. It has a ASMedia 1064 chip.
https://www.amazon.fr/gp/product/B08NXTF7BM/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
I have been using it in some salvaged gear for like 3 years without issues, then I upgraded my hardware and hit this issue:
I used another computer to upgrade the firmware (by following the guide, thanks PhilBarker! https://docs.phil-barker.com/posts/upgrading-ASM1166-firmware-for-unraid/ ).
And now it works with my new hardware.
I've read that some chips may prevent the CPU going to deep C-states, I hadn't time to experiment with this yet.