The variety-invariance conjecture for Bol–Moufang homology

Let $Vij$ and $Wk\ell$ be identities defining the same variety of quasigroups, let $X$ be any quasigroup in that variety, and let $(t,s)$ be a substitution. Write $H_2(X;Vij,t,s)$ for the second Bol–Moufang homology associated with the identity $Vij$ and substitution $(t,s)$. Variety-invariance conjecture. The homology depends only on the variety containing $X$ and on $(t,s)$; specifically, for all $(t,s)$,

$$H_2(X;Vij,t,s)=H_2(X;Wk\ell,t,s).$$

Moreover, when $X$ is a group, any identity $Vij$ defining the variety of groups yields the usual group homology of $X$. The authors verify independence from the particular identity in their examples, but state that the general claim has not been proved; the conjecture also proposes agreement with ordinary group homology in the group case.