Mu–Welker's total positivity conjecture for the barycentric subdivision transformation matrix

Let $\mathcal{F}$ be the barycentric subdivision, and let $H_{\mathcal{F}}$ be the transformation matrix encoding the transformation of the $h$-vector under an $\mathcal{F}$-uniform subdivision. A matrix is totally positive (TP) if all its minors are non-negative. Mu–Welker's conjecture. The transformation matrix $H_{\mathcal{F}}$ is TP. Mu and Welker had shown that this matrix is $\mathrm{TP}_2$, meaning that all minors of orders at most $2$ are non-negative; the conjecture asks for non-negativity of minors of every order.