Optimality of extremely branched trees for the Basic regression

Let $\mathcal{R}(n)$ be the class of trees on $n$ vertices, and let $BP^0$ denote the Basic regression with minimum value $BP^{0*}_\mathcal{R}(n)$ over $\mathcal{R}(n)$. A tree is pendent-rooted when its root is a pendent vertex, and an extremely branched tree is the tree type depicted in the paper's minimizing-tree figure. Extremely branched tree conjecture. If

$$BP^0(T)=BP^{0*}_\mathcal{R}(n)$$

for some pendent-rooted tree $T\in\mathcal{R}(n)$, then $T$ is an extremely branched tree. The claim formalizes the paper's hypothesis that an extremely branched tree optimizes the Basic regression, although the authors explain that they cannot prove this from the available inequalities for the ad hoc index $C^0$.