Higher-order Eulerian triangle total positivity conjecture

For $r\geq 1$, let $E^{(r)} = \left(\genfrac{\langle}{\rangle}{0pt}{}{n}{k}^{\! (r)}\right)_{n,k\geq 0}$ be the $r$th-order Eulerian triangle, and let $\widecheck{E}^{(r)} = \left(\genfrac{\langle}{\rangle}{0pt}{}{n}{n-k-1}^{\! (r)}\right)_{n,k\geq 0}$ be its shifted reversal. Higher-order Eulerian conjecture. Both $E^{(r)}$ and $\widecheck{E}^{(r)}$ are totally positive for every $r\geq 1$. The cases concerning real-rootedness and coefficientwise Hankel-total positivity are already known, but the supplied status evidence does not establish that the two total-positivity assertions themselves are solved.