Conjecture on abelianized monodromy invariants for symplectic branched covers

Let $(X,\omega)$ be a simply connected symplectic $4$-manifold. For the degree-$N_k$ symplectic branched covering associated with sufficiently large $k$, let $G_k^0$ be the relevant subgroup of the covering complement group, let $\Lambda_k\subset\mathbb{Z}^2$ be the subgroup determined by the numerical canonical and hyperplane classes, and let

$$\phi_k:G_k^0\longrightarrow (\mathbb{Z}^2/\Lambda_k)^{N_k-1}$$

be the homomorphism defined by the homology classes of lifts of loops. Abelianized monodromy conjecture. For all large enough $k$, $\phi_k$ induces an isomorphism

$$\operatorname{Ab}G_k^0\simeq (\mathbb{Z}^2/\Lambda_k)^{N_k-1}.$$

The conjecture extends the corresponding theorem known for the listed computable examples in class $(\mathcal{C})$. It asserts the expected abelianized structure for every simply connected symplectic $4$-manifold; the source gives no resolution.