Conjecture on log-concavity of the signed binomial-Eulerian polynomials

For $n\geq1$, let $\tilde{A}_n(t,q)$ denote the $q$-binomial-Eulerian polynomial, and specialize it at $q=-1$ to obtain the signed binomial-Eulerian polynomial $\tilde{A}_n(t,-1)$. A polynomial $h(t)=\sum_{k=0}^na_k t^k$ is log-concave when its coefficients satisfy $a_k^2-a_{k-1}a_{k+1}\geq0$ for the relevant indices $k$.

The signed log-concavity conjecture. The polynomial $\tilde{A}_n(t,-1)$ is log-concave for $n\geq1$.

This is the second conjecture proposed in the closing remarks. The supplied text gives no evidence of a resolution, so its status remains open.