Automorphic and geometric equivalence in subvarieties of groups

Let $\Theta$ be a subvariety of the variety of all groups. In this setting, automorphic equivalence and geometric equivalence are the two equivalence relations on algebras in $\Theta$ considered in the source.

Automorphic–geometric equivalence conjecture. In every subvariety $\Theta$ of the variety of all groups, automorphic equivalence coincides with geometric equivalence.

The conjecture would extend the known coincidence for the variety of all groups, the variety of all abelian groups, and the varieties of nilpotent groups of fixed class, while addressing the possibility of nontrivial groups $\mathfrak{A/Y}$ in other group varieties. The source reports no resolution.