Dwyer quotient conjecture for the Grigorchuk group

Let ${\mathfrak G}$ be the Grigorchuk group, and let $M_c({\mathfrak G})$ denote its $c$-th Dwyer quotient. Dwyer quotient conjecture.

$$M_c({\mathfrak G}) \cong \begin{cases} {\mathbb Z}_2, & c=1,\\ ({\mathbb Z}_2)^2, & c=2,\\ ({\mathbb Z}_2)^{2m+3}, & c\in\{3\cdot 2^m,\ldots,3\cdot 2^{m+1}-1\}, \end{cases}$$

with $m\in{\mathbb N}_0$. The conjecture predicts the complete pattern of the computed Dwyer quotients and would in particular describe the infinitely generated Schur multiplier of the group.