Spin and characteristic-class conjecture for the commutator subgroup

Let $X$ be a simply connected symplectic manifold, let $G_k^0(X,\omega)$ be the reduced stabilized fundamental group, and let $L_k$ and $K_X$ be the classes appearing in the construction. Define $\Gamma_1$ by $\Gamma_1=\mathbb{Z}_2$ if $X$ is spin and $\Gamma_1=1$ otherwise, and define $\Gamma_2$ by $\Gamma_2=\mathbb{Z}_2$ if $L_k\equiv K_X\pmod 2$ and $\Gamma_2=1$ otherwise. Commutator subgroup structure conjecture. If $k$ is large enough, then

$$[G_k^0,G_k^0]\simeq\Gamma_1\times\Gamma_2.$$

This proposed refinement is based on a relatively small amount of evidence from the examples in the paper and remains open.