Dwyer quotient conjecture for the twisted twin of the Grigorchuk group

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

$$M_c(\bar{\mathfrak G}) \cong \begin{cases} ({\mathbb Z}_2)^2,({\mathbb Z}_2)^5,\text{ or }({\mathbb Z}_2)^7, & c=1, c=2,\text{ or }c=3,\text{ respectively},\\ ({\mathbb Z}_2)^{4(m+1)+4}, & c\in\{2^{m+2},\ldots,2^{m+2}+2^{m+1}-1\},\\ ({\mathbb Z}_2)^{4(m+1)+7}, & c\in\{2^{m+2}+2^{m+1},\ldots,2^{m+3}-1\}, \end{cases}$$

with $m\in{\mathbb N}_0$. This conjecture records the periodic rank pattern observed in computations of the Dwyer quotients.