Elementary equivalence conjecture for finite-group coset expansions of non-abelian free groups

Let $Q$ be a finite group, let $F_{1}$ and $F_{2}$ be non-abelian free groups, and for $i=1,2$ let $\pi_i:F_i\twoheadrightarrow Q$ be epimorphisms. Let $\mathcal{L}^{Q}$ be the language of groups expanded by unary predicates $\{P_q\}_{q\in Q}$, where in $(F_i,\pi_i)$ the predicate $P_q$ is interpreted as $\pi_i^{-1}(q)$. Elementary equivalence conjecture. The pairs $(F_{1},\pi_{1})$ and $(F_{2},\pi_{2})$ are elementarily equivalent as $\mathcal{L}^{Q}$-structures. This conjecture proposes that all such finite-group coset expansions of non-abelian free groups have the same first-order theory, extending the elementary-equivalence phenomenon for non-abelian free groups and the model-theoretic setting underlying Sela's work. The supplied text gives no resolution, so its status remains open.