Abelianization conjecture for braid-group monodromy of symplectic branched covers

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

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

be the homomorphism described through lifts of loops. Abelianization conjecture. For all sufficiently large $k$, the homomorphism $\phi_k$ induces an isomorphism

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

This conjecture extends the corresponding theorem known for the listed computable examples in class $(\mathcal{C})$. It leaves open the general simply connected symplectic $4$-manifold case; unlike the theorem, it makes no assertion here about $\operatorname{Ker}\phi_k$.