Finite Borel asymptotic dimension of bounded-to-one commutative monoid actions

We prove that every bounded-to-one action of a finitely generated commutative monoid has finite asymptotic dimension, without a freeness assumption. If the group completion has torsion-free rank $r$, we give an explicit upper bound $(3^{r+1}-3)\mathrm{/}2$. This answers a question of Shinko, Weilacher and Yu, and extends a theorem of theirs.