The one-ended subgroup characterization for inaccessible finitely generated groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. Say that $H$ is 1-ended in $G$ when it has one end relative to $G$.

The one-ended subgroup characterization. The subgroup $H$ of the finitely generated group $G$ is 1-ended in $G$ if and only if for each splitting $A\ast_C B$ of $G$, where $A\ne C\ne B$ and $C$ is finite, $H$ is a subgroup of a conjugate of $A$ or $B$.

This question is raised in the context of inaccessible finitely generated groups; the paper asks what can be said about such groups. The source provides no resolution of the characterization.