Optimal Steiner tree construction conjecture for regular simplices

Let $d$ be the number of terminals in a regular simplex, and let $k\geq 1$. The construction described immediately before the claim recursively splits Steiner points and coordinates from a known Steiner tree. Optimal construction conjecture. This construction yields an optimal Steiner tree for every regular $d$-simplex when $d=2^k$, and its natural generalization yields the optimal Steiner tree for every regular $d$-simplex with $d\geq 3$. The source gives no resolution for the asserted generalization.