The undecidability conjecture for infinite stature in graphs of free groups

Let $G$ be a group that splits as a finite graph of finitely generated free groups. An algorithm is said to certify that $G$ has infinite stature with respect to its vertex groups if, on input $G$, it outputs such a certification. Undecidability conjecture. There is no algorithm that takes as input $G$ and outputs a certification that $G$ has infinite stature with respect to its vertex groups. This is posed as an expected consequence after a proposition characterizing finite stature for graphs of groups with finite-index edge groups; the source provides no resolution or supporting evidence beyond this expectation.