Steiner distance Graham–Pollak determinant conjecture for trees

Let $D_k(T)$ denote the Steiner distance hypermatrix of order $k$ associated with a tree $T$ on $n$ vertices. Steiner distance Graham–Pollak determinant conjecture. The quantity $\det(D_k(T))$ is a function only of $n$ and $k$ for trees $T$ on $n$ vertices. The claim is presented as a full generalization of the Graham–Pollak Tree Theorem for Steiner distance. It is stated to hold trivially for $n\leq 3$ or $k$ odd, and has been checked computationally for $(k,n)=(4,4)$, $(4,5)$, and $(6,4)$; the general assertion remains open.