Abelianization and commutator conjecture for stabilized fundamental groups

Let $X$ be a simply connected symplectic manifold, let $G_k^0(X,\omega)$ be the reduced stabilized fundamental group, let $\Lambda_k$ be the image of $\lambda_k:H_2(X,\mathbb{Z})\to\mathbb{Z}^2$, and let $n_k=\deg f_k$. Let $\mathcal{R}_{n_k}$ denote the reduced regular representation of $S_{n_k}$, so that $\mathcal{R}_{n_k}\simeq\mathbb{Z}^{n_k-1}$. Abelianization and commutator conjecture. If $k$ is large enough, then

$$\operatorname{Ab}G_k^0(X,\omega)\simeq (\mathbb{Z}^2/\Lambda_k)\otimes\mathcal{R}_{n_k},$$

and the commutator subgroup $[G_k^0,G_k^0]$ is a quotient of $(\mathbb{Z}_2)^2$. The theorem preceding this conjecture supplies a natural surjection onto the displayed abelian group; the conjecture predicts injectivity and gives a bound on the commutator subgroup, supported by the examples discussed in the paper.