Persistent containment conjecture for finitely generated residually finite groups

Let $H$ be a finitely generated residually finite group. A group $G$ persistently contains $H$ if every finite-index subgroup $G'\subset G$ contains a subgroup isomorphic to $H$. Persistent containment conjecture. There exists a finitely generated residually finite group $G$ that persistently contains $H$. This would extend the paper's theorem, which establishes the analogous assertion for $H$ equal to Promislow's group, and the general case remains open in the source.