Relative Boone--Higman conjecture

A normal pair is a pair $(G,N)$ consisting of a group and a normal subgroup. A group $\Gamma$ sharply embeds in $(G,N)$ when the normal pair $(\Gamma,\{1\})$ embeds in $(G,N)$; a group is relatively simple when it has a largest normal subgroup in the sense defined in the source.

Relative Boone--Higman conjecture. Let $\Gamma$ be a finitely generated group with solvable word problem. Then $\Gamma$ sharply embeds in a finitely presented relatively simple group.

This is a relative strengthening of the Boone--Higman conjecture. The source relates it to embeddings that intersect every proper normal subgroup trivially and leaves the conjecture open.