Conjecture on stabilized fundamental groups of branch curve complements

Let $X$ be a simply connected algebraic surface, let $D\subset\mathbb{CP}^2$ be the branch curve of the relevant projection, and let $G^0$, $K$, $\bar{G}$, $\phi$, $\Lambda$, and $N$ have the meanings determined by the symplectic stabilization construction. For sufficiently large $k$, the following should hold. Stabilized fundamental-group conjecture. The symplectic stabilization operation is trivial, namely

$$K=\{1\},\qquad \bar{G}=\pi_1(\mathbb{CP}^2-D).$$

Moreover, the homomorphism

$$\phi:\operatorname{Ab}(G^0)\to (\mathbb{Z}^2/\Lambda)^{N-1}$$

is an isomorphism, and the commutator subgroup satisfies

$$[G^0,G^0]\text{ is a quotient of }\mathbb{Z}_2\times\mathbb{Z}_2.$$

The conjecture summarizes the simple structure observed in the known examples of sufficiently high-degree projections, including projections of simply connected algebraic surfaces. Its status is not resolved by the supplied text.