Non-coprime-power obstruction to tree-pair representations of Bieri–Strebel groups

Let $k>1$, let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be coefficients that are not all zero. Let the Bieri–Strebel group have associated subdivision polynomial

$$a_nx^{kn}+a_{n-1}x^{k(n-1)}+\cdots+a_1x^k-1.$$

Non-coprime-power conjecture. This Bieri–Strebel group does not have a well-defined tree-pair representation.

The conjecture generalizes the preceding obstruction for subdivision polynomials involving non-coprime powers. The source presents it as a proposed generalization after proving the relevant quartic case, but gives no resolution of the general claim.