Solvable-word-problem preformation conjecture

Let $G$ be an infinite group with solvable word problem and rank $k$. Say that $G$ preforms $H$ when $H$ is a limit of marked copies of $G$ in the space of marked groups.

Solvable-word-problem preformation conjecture. There exists a group $H$, non-isomorphic to $G$, with solvable word problem and rank at most $k+1$, such that $G$ preforms $H$.

The statement is motivated by examples such as the Grigorchuk group preforming a free group. The paper explicitly says that it is believed to hold in general but cannot currently be proved.