Hyperfiniteness of bounded-to-one actions of commutative monoids

A theorem of Dougherty–Jackson–Kechris states that any equivalence relation generated by a single Borel function is hypersmooth. A well-known open problem is whether this can be generalized to equivalence relations generated by countable families of pairwise commuting Borel functions. We give an affirmative answer in the case where the functions are bounded-to-one. This generalizes the theorem of Gao–Jackson on Borel actions of countable abelian groups.