Relative permutational Boone--Higman conjecture

Let $G\curvearrowright S$ be an action, let $\ker(G\curvearrowright S)$ denote its kernel, and let $SV_G$ be the associated twisted Brin--Thompson group. A group $\Gamma$ sharply embeds in $(G,N)$ when $(\Gamma,\{1\})$ embeds in the normal pair $(G,N)$.

Relative permutational Boone--Higman conjecture. Let $\Gamma$ be a finitely generated group with solvable word problem. Then there exists a group $G$ with a type $[\operatorname{A}_2]$ action on a set $S$ such that $\Gamma$ sharply embeds in $(G,\ker(G\curvearrowright S))$, and hence $\Gamma$ sharply embeds in the finitely presented relatively simple group $SV_G$.

This combines the relative embedding condition with the permutation-action formulation. It is presented as a conjectural strengthening related to the classical and relative Boone--Higman conjectures.