Total positivity of the barycentric subdivision operator

Let $\mathcal{F}$ be the barycentric subdivision, and let $H_{\mathcal{F}}$ denote its associated operator. A matrix is totally positive if every minor of every order is positive, and $H_{\mathcal{F}}$ is TP when it is totally positive. Total-positivity conjecture. The matrix $H_{\mathcal{F}}$ is TP. This would strengthen the known result that $H_{\mathcal{F}}$ is TP$_2$, and would extend the corresponding theorem for the $r$th-edgewise subdivision; the source offers experimental evidence but does not report a proof.