Reversed higher-order quasi-Eulerian triangle total positivity conjecture

For $r\geq 2$, let $Q_{\rm C}^{(r)}$ and $Q_{\rm S}^{(r)}$ be the quasi-Eulerian cycle and subset triangles, respectively, and let $M^{\rm rev}$ denote reversal of the rows of a matrix $M$. Quasi-Eulerian reversal conjecture. The matrices $\left(Q_{\rm C}^{(r)}\right)^{\rm rev}$ and $\left(Q_{\rm S}^{(r)}\right)^{\rm rev}$ are totally positive for all $r\geq 2$. These assertions were verified computationally for $3\leq r\leq 10$ on leading $50\times 50$ submatrices, but remain open in general.