Real-rootedness conjecture for the squares of the Eulerian and Delannoy triangles

Let $A$ and $D$ denote the Eulerian and Delannoy triangles, respectively, and let $A^2$ and $D^2$ be their matrix squares. For a lower triangular matrix $M$, write its $n$-th row generating function as $M_n(x)=\sum_k M(n,k)x^k$. Real-rootedness conjecture. The row generating functions of $A^2$ and $D^2$ have only real nonpositive roots. Numerical evidence supports this assertion for the row generating functions of $D^2$, while the conjecture concerns both triangles and remains open in the supplied text.