Unbounded vertices conjecture for arbitrary-tree phylogenetic polytopes

Let $T$ be a rooted tree, let ${\mathcal A}_T$ be the configuration of exponent vectors from the toric parametrization, and let $P_T$ be the convex hull of the columns of ${\mathcal A}_T$. As $T$ ranges over arbitrary trees, the number of vertices of $P_T$ should be unbounded. The paper contrasts this with the conjectured boundedness for binary trees and supports the assertion with computations, including a 15-node tree whose polytope has 34 vertices; a general proof is not supplied.