Galois-cover fundamental group conjecture

Let $X$ be a simply connected complex surface, let $f_k:X\to\mathbb{CP}^2$ be a generic projection defined by sections of $L^{\otimes k}$, let $\tilde{X}_k$ be the associated Galois cover, let $L_k$ denote the corresponding class, and let $n_k=\deg f_k$. Galois-cover fundamental group conjecture. For sufficiently large $k$,

$$\pi_1(\tilde{X}_k)\simeq (\mathbb{Z}_{ks})^{n_k-2},$$

where $ks$ is the divisibility of $L_k$ in $H_2(X,\mathbb{Z})$. This predicts an explicit fundamental group for Galois covers associated with generic projections; it is stated as holding in the examples treated in the paper, but the general assertion remains open.