Infinite-order monodromy for hypersurface singularities with positive geometric genus

Prove that for every 2-dimensional weighted-homogeneous isolated hypersurface singularity V = {f = 0} whose link Y is a rational homology 3-sphere and whose geometric genus satisfies p_g(V) > 0, the monodromy ψ of the Milnor fibration has infinite order in the smooth mapping class group MCG(M) of a Milnor fiber M.

Background

The paper studies the smooth mapping class group behavior of monodromies of Milnor fibrations for 2-dimensional isolated hypersurface singularities. For ADE (p_g = 0) singularities, simultaneous resolution implies the monodromy cannot have infinite order in MCG(M), whereas the authors prove affirmative results for minimally elliptic (p_g = 1) cases.

Motivated by these findings, the authors conjecture that the failure of simultaneous resolution for p_g > 0 should have a smooth-topological explanation: namely, that the smooth monodromy is of infinite order. This conjecture is stated as a special case of a broader conjecture formulated later for general isolated surface singularities.