Optimal topology conjecture for Steiner trees of the regular simplex

Let $k\geq 1$ and let $d$ satisfy

$$2\leq 2^k<d\leq 2^{k+1}.$$

A good binary tree of height $k$ is defined recursively: a height-zero tree is a single node, and for positive height at most one child-subtree is good of height $k-1$, while the remaining child-subtrees are full binary trees of height $k-1$ or $k-2$. Optimal topology conjecture. The topology of the optimal Steiner tree of a regular $d$-simplex is formed by taking the good tree of height $k+1$ on $d$ leaf nodes, removing its root node, and reconnecting the tree by an edge between the former children of the root. This is stated as a reformulation of a conjecture of Smith. The source does not provide a resolution.