S-integral preperiodic points for monomial semigroups over number fields

We consider semigroup dynamical systems defined by several monnomials over a number field K. We prove a finiteness result for preperiodic points of such systems which are S-integral with respect to a non-preperiodic point β, which is uniform as β varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap, and verifies a special case of a natural generalisation of a conjecture of Ih.