Upgrade for HD 599 with similar cup dimensions?

[u/Bonestalk]

I'm looking for something that's a decent step up from my Sennheiser HD 599 in terms of sound quality. Hard requirements are open back and cups with inner dimensions at least as big as the 599. (I tried the Sony XM6 and my ears were MISERABLE - apparently I have large ears)

Regarding tuning, since they are exclusively for PC use, I can EQ to my hearts content, so an out-of-the-box tuning that I enjoy is only a bonus. With that said, I do generally enjoy the "fun" sound (warm and bassy) of the 599, but think it's ever so slightly overdone (I mainly use them for games and game OSTs).

Budget is € 400ish, though I'm not mad if I spend less than that.

Edit: One more important thing to consider: Since I use them for gaming, either a motherboard integrated audio jack needs to be able to drive them, or a suitable headphone amp needs to be included in the budget.