On what codec settings?. IMO h256 and AV1 are extremely close depending on individual, and still experimental, codec tweaks.
MP3 coding was near garbage at 128kbps, but over the years the codec community managed to make 128kbps significantly better audibly without changing the format
Next gen h256 coders will likely copy whatever non-bitstream optimisation they can from AV1 coders, and they will be a near match again.
AV1 beats h265 by some 10-15% and it's still improving. h265 has had plenty of time to mature. I don't think there's much headroom for h265 and anyway the JVET/MPEG has moved on to VVC, which beats AV1 by some 25-29% and it is still improving. They know they'll lose the short term war and are looking longer term.
AV1 was supposed to be "on par" with h265 but has jumped ahead by a bit and will likely improve for quite awhile until AV2 is worked on. Will it reach VVC levels? Probably not. But it will certainly be better than h265/HVEC.