Conjecture on the real-rootedness and strong log-concavity of the Birkhoff-polytope h-star polynomial

Let $h_n(t)$ be the Ehrhart polynomial associated with the $n$-th Birkhoff polytope, and define

$$N=\left\lfloor\frac{(n-1)^2}{4}\right\rfloor,\qquad \hat N=\left\lfloor\frac{(n-2)^2}{2}\right\rfloor.$$

Define its normalized generating polynomial by

$$h_n^*(y)=(1-y)^{(n-1)^2+1}\sum_{t\ge0}h_n(t)y^t.$$

The real-rootedness and log-concavity conjecture. One has

$$h_n^*(y)=y^N\sum_{i=0}^{\hat N}a_i y^i,$$

where the $a_i$ are positive integers; moreover, $h_n^*(y)$ is real-rooted, and $(a_0,a_1,\dots,a_{\hat N})$ is palindromic, unimodal, and strongly log-concave. Here strong log-concavity means that, after setting $b_i=a_i/\binom{\hat N}{i}$, one has $b_i^2\ge b_{i-1}b_{i+1}$ for $1\le i\le\hat N-1$. These properties are proposed as analogues of known structural properties of Ehrhart-series numerators; the source gives no resolution status.