The presentation conjecture for the subgroup H_k of G_{k+1}^k

Let $G_{k+1}^k$ be one of the groups under consideration, and let $H_k$ be its subgroup generated by $b_1,\dots,b_k$. For distinct indices $i,j,m,l\in\{1,\dots,k\}$, consider the relations $b_1^2=\dots=b_k^2=1$ and $[(b_i b_j)^2,(b_m b_l)^2]=1$. Presentation conjecture. For any group $G_{k+1}^k$, its subgroup $H_k$ admits the presentation

$$H_k=\langle b_1,\dots,b_k\mid b_1^2=\dots=b_k^2=1,\ [(b_i b_j)^2,(b_m b_l)^2]=1\rangle,$$

where $\{i,j,m,l\}$ ranges over all possible sets of distinct numbers from $\{1,\dots,k\}$. This would provide a uniform presentation useful for studying the word problem in the groups $G_{k+1}^k$, which is known in the cases treated earlier in the paper but is not established here for arbitrary $k$.