Sign conjecture for Steiner distance hyperdeterminants

Let $T$ be a tree on $n$ vertices, let $M$ be its order-$k$ Steiner distance hypermatrix, and write $\operatorname{sgn}(x)$ for $1$ if $x>0$, $-1$ if $x<0$, and $0$ if $x=0$.

Sign conjecture. For any tree on $n$ vertices,

$$\operatorname{sgn}(\det(M))=(-1)^{n-1}$$

when this quantity is nonzero.

The conjecture appears in earlier work and has been checked numerically for $(k,n)=(4,4)$, $(4,5)$, and $(6,4)$. It is trivially true for $n\leq 3$ and odd $k$, and follows from Graham–Pollak when $k=2$; its general status is not established in the supplied text.