The branch-length conjecture for pods of conjugates in the Bass–Serre tree

Let $f$ be the polynomial automorphism under consideration, and let an $n$-pod be a configuration formed by $n$ conjugates of $f$ in the Bass–Serre tree, with consecutive branches understood cyclically as in the surrounding construction. Branch-length conjecture. If $n$ conjugates of $f$ form an $n$-pod in the Bass–Serre tree, with two consecutive branches of length $4$, then

$$n \geq 6.$$

This condition would ensure that the curvature of the corresponding triangular region is non-positive and is intended to extend the paper's strategy to automorphisms of length $12$, where triangles with three edges of length $4$ arise. Whether the assertion holds is left as a problem for extending the results.