Monster Hunter Rise has just released on Steam today from Capcom and the good news is - it appears to run very nicely out of the box with Steam Play Proton on Linux. That's another tick in the box for a big AAA title.
Tested with Proton Experimental, the only issue currently encountered is a small intro video not playing. This is a reoccurring issue and will be for the Steam Deck, for titles that use things like Media Foundation. If such things bother you, it worked just fine with Proton GE which you can easily download with ProtonUp-Qt. Over time though, Valve will be re-encoding videos for games where it's an issue so eventually more won't see this issue. Luckily here it's really minor.
I haven't been able to see any obvious issues, there isn't even much stutter which you would usually see from a brand new release on Linux. Then again, it's not as a graphically intense as certain other big releases, since this was originally developed for the Nintendo Switch so it's really not all that surprising.
Overall, a pretty smooth release to be able to play at release on Linux with Proton.