The strong K-two-threes conjecture for weighted graph distance matrices

Let $G$ be a graph, let $\phi$ be an edge-weighting, and let $D_{\phi G}$ denote the weighted distance matrix of $G$. For a positive integer $n$, let $K_{2,3,\ldots,3}$ denote the complete multipartite graph with one part of size $2$ and $n$ parts of size $3$. The strong K-two-threes conjecture. The matrix $D_{\phi G}$ has at most $n$ positive eigenvalues for every edge-weighting $\phi$ if and only if $G$ has no $K_{2,3,\ldots,3}$ minor, with $n$ parts of size $3$. This is presented as a stronger and more general form of the weak conjecture, motivated by the stated inertia of the unweighted graph $K_{2,3,\ldots,3}$; the source does not report a resolution.