The simplicial freeness conjecture for the Brunnian planar braid subgroup

Let $SPT_*$ be the simplicial group of simplicial pure twin groups, let $c_{111}$ be the 2-simplex with trivial faces, and let $K_*$ be the smallest simplicial subgroup of $SPT_*$ containing $c_{111}$. Let $S^2$ be the simplicial sphere with non-degenerate 2-simplex $\sigma=(0,1,2)$, and let $F[S^2]_*$ be Milnor's free simplicial group on $S^2$. The simplicial homomorphism

$$\Theta:F[S^2]_*\longrightarrow K_*$$

is induced by the simplicial map sending $\sigma$ to $c_{111}$. The simplicial freeness conjecture. The homomorphism

$$\Theta:F[S^2]_*\longrightarrow K_*$$

is an isomorphism. This would identify the simplicial subgroup generated by $c_{111}$ with Milnor's free simplicial group on the simplicial 2-sphere, extending the verified injectivity in degrees at most $4$ and providing a description of the homotopy groups of $S^3$ through the Moore complex of $K_*$.