The splitting conjecture for finitely presented non-semistable groups

Let $G$ be a finitely presented group. Say that $G$ splits non-trivially when it admits a nontrivial splitting as a graph-of-groups decomposition.

The splitting conjecture for finitely presented non-semistable groups. If $G$ does not have semistable fundamental group at infinity, then $G$ splits non-trivially.

The conjecture is presented as a prospective converse to combination results showing that suitable finite graphs of groups with semistable vertex groups have semistable fundamental group at infinity. The source provides no resolution of the converse.