TBR adjacency implies BME adjacency

Let $\mathcal{T}_n$ be the space of trees on $n$ leaves. For $T,T'\in\mathcal{T}_n$, let $d_{TBR}(T,T')$ denote the minimum number of tree-bisection-regrafting moves needed to transform $T$ to $T'$. Let $d_{BME}(T,T')=1$ when the corresponding vertices $\mathbf{w}^T$ and $\mathbf{w}^{T'}$ are joined by an edge in the balanced minimum evolution polytope $\mathcal{P}_n$. TBR adjacency conjecture. If $T,T'\in\mathcal{T}_n$, then

$$d_{TBR}(T,T')=1$$

implies

$$d_{BME}(T,T')=1.$$

Nearest-neighbor interchange adjacency implies subtree-prune-regraft adjacency, and subtree-prune-regraft adjacency is known to imply BME adjacency. The paper reports no examples showing that TBR adjacency fails to imply BME adjacency, so the conjecture remains open.