Steiner distance hyperdeterminant conjecture for trees

Let $T$ be a tree on $n eq 3$ vertices, let $k$ be a positive integer, and let $d4ae_T$ be the order-$k$ Steiner distance hypermatrix of $T$, whose $(v_1,d7e2,d7e3,d7e4,d7e5,d7e6)$ entry is the Steiner distance $d_T(v_1,d7e2,d7e3,d7e4,d7e5,d7e6)$. Its hyperdeterminant is said to depend on $T$ only through $n$, and to vanish exactly when $k$ is odd. Steiner distance hyperdeterminant conjecture. The order-$k$ Steiner distance hypermatrix of a tree $T$ on $n e 3$ vertices has a hyperdeterminant that depends on $T$ only through $n$, and this hyperdeterminant is $0$ if and only if $k$ is odd. The surrounding text says that the paper proves results for odd-order Steiner hypermatrices and reports computation suggesting extension to even order, so the full asserted statement is not established in the supplied material.