Infinite logconcavity conjecture for the polynomials $A_n(x)$

Let $A_n(x)$ be the polynomial defined earlier in the paper, and let a sequence be infinite logconcave when every iterate of the operator $\mathfrak{L}$ produces a positive sequence. Infinite logconcavity conjecture. For every $n \in \mathbb{N}$, the polynomial $A_n(x)$ is infinite logconcave. This is supported by symbolic calculations and was verified in the paper for all $n \leq 300$; the assertion for arbitrary $n$ remains open.