Recognizability conjecture for marked isolated groups

Let $\mathcal{G}$ be the space of marked groups and let $\mathcal{G}_{WP}$ be its word-problem subspace. A marked group is $\Lambda_{WP}$-recognizable when its singleton is decidable relative to the word-problem representation; an isolated marked group is a singleton open subset of $\mathcal{G}$.

Marked-isolated-groups conjecture. Marked isolated groups are the only marked groups that can be recognized for $\Lambda_{WP}$; equivalently, they are the only marked groups whose singleton subsets of $\mathcal{G}$ are $\Lambda_{WP}$-decidable.

Isolated marked groups are known to be recognizable, so the conjecture concerns the converse. The paper notes that whether a group with solvable word problem can be isolated in $\mathcal{G}_{WP}$ without being isolated in $\mathcal{G}$ remains open.