Strict total positivity conjecture for the Eulerian square

Let $A=[A(n,k)]_{n,k\geq 1}$ be the Eulerian triangle, where $A(n,k)$ is the Eulerian number, and define its square by

$$A^{\ulcorner}=\left[A(n+k,k)\right].$$

Eulerian-square conjecture. The Eulerian square $A^{\ulcorner}$ is strictly totally positive: all its minors are positive. The conjecture is motivated in the source by the strict total positivity of the Narayana squares, and no resolution is supplied.