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Aye that’s a good point, one of the jockey wheels has a narrow wide profile. fwiw I tested a Rival AXS mech on a mountain bike with an 11 speed chain and cassette and it shifted fine in the work stand.
The eagle axs jockey wheels mesh fine with a 12 speed shimano mtb chain.
There’s no need to use an eagle chain though, 12 speed shimano road and mtb chains are the same.Having said all that, I think it’s very likely the Rival mech would work fine without bothering to swap out one of the jockey wheels.
IIRC the Rival AXS RD jockey wheels (or at least one of the two) have a narrow-wide profile specific for the flat-top chain (or its bigger rollers, to be precise). Some people reported getting around this (i.e. avoiding having to use the flat-top chain) by buying a set of GX AXS jockey wheels and swapping them, then use an Eagle chain. Not sure if the Shimano 12-speed chain is different and works with the stock Rival AXS pulley wheels. I wonder if it was possible to just do some dremeling to modify the (potentially) incompatible Rival pulley wheel.