Gromov's surface subgroup conjecture for one-ended hyperbolic groups

Let $G$ be a one-ended word-hyperbolic group, meaning a word-hyperbolic group with one end. Let the surface group of genus $2$ be the fundamental group of a closed orientable surface of genus $2$. Gromov's conjecture. The group $G$ contains an embedded copy of the surface group of genus $2$. If true, together with the inheritance of domino-problem undecidability by groups containing suitable subgroups, this would imply undecidability for every one-ended word-hyperbolic group. The conjecture is presented in the source without a resolution.