Fel'shtyn–Troitsky conjecture on property $R_\infty$ and solvable-by-finite groups

Let $G$ be a finitely generated residually finite group. Fel'shtyn–Troitsky conjecture. The group $G$ either has property $R_\infty$ or is solvable-by-finite. The disjunction is not exclusive: some solvable-by-finite groups also have property $R_\infty$. The conjecture was proved for all groups with finite upper (Prüfer) rank, but remains open in general.