Toroidalization of generating sequences in dimension two function fields

Let k be an algebraically closed field of characteristic 0, and let K∗/K be a finite extension of algebraic function fields of transcendence degree 2 over k. Let ν∗ be a k-valuation of K∗ with valuation ring V∗, and let ν be the restriction of ν∗ to K. Suppose that R→S is an extension of algebraic regular local rings with quotient fields K and K∗ respectively, such that V∗ dominates S and S dominates R. We prove that there exist sequences of quadratic transforms and along ν∗ such that dominates and the map between generating sequences of ν and ν∗ has a toroidal structure. Our result extends the Strong Monomialization theorem of Cutkosky and Piltant.