The non-amenable residually finite property $R_\infty$ conjecture

Let $G$ be a finitely generated residually finite non-amenable group. Non-amenable property $R_\infty$ conjecture. The group $G$ has property $R_\infty$. This is presented as a weaker variant of the preceding conjecture, because solvable-by-finite groups are amenable. Its status is not resolved in the supplied text.