Infinite-order monodromy conjecture for positive-geometric-genus surface singularities

Let $V = \{f = 0\}$ be a $2$-dimensional weighted-homogeneous isolated hypersurface singularity with rational homology $3$-sphere link $Y$, Milnor fiber $M$, and monodromy $\psi$ of its Milnor fibration. The geometric genus of $V$ is denoted by $p_g(V)$. Infinite-order monodromy conjecture. The answer to whether $\psi$ has infinite order in the smooth mapping class group $\operatorname{MCG}(M)$ is affirmative when $p_g(V)>0$. The claim extends the known affirmative result for $p_g(V)=1$ and contrasts with the negative answer for ADE singularities, which have $p_g(V)=0$; the general case remains open.