The monotone total positivity conjecture for the Eulerian matrix

Let $A(n,k)$ be the Eulerian number, and let $A=(A(n+1,k))_{n,k\in\mathbb{N}}$ be the Eulerian matrix. For $i,j,r\in\mathbb{N}$, take the submatrix whose rows are indexed by $i,i+1,\ldots,i+r$ and whose columns are indexed by $j,j+1,\ldots,j+r$.

Monotone total positivity conjecture. For all $i,j,r\in\mathbb{N}$, the determinant of this submatrix is a monotonically increasing function of $i\in\mathbb{N}$.

This is proposed as a stronger form of total positivity for the Eulerian matrix. The supplied text gives no resolution, so the assertion remains open.