Real-rootedness conjecture for Narayana polynomials over binary nestohedra

For a positive integer $n$, define the Narayana polynomial

$$N_n(x)=\sum_{i=1}^n\frac{1}{n}\binom{n}{i}\binom{n}{i-1}x^i.$$

Given a composition $\mathbf q=(q_0,\ldots,q_k)\in\mathbb Z_{>0}^{k+1}$, define

$$N_{\mathbf q}(x)=\prod_{i=0}^kN_{q_i}(x).$$

An integral binary nestohedron $P\subseteq\mathbb R^{k+1}$ is a binary nestohedron whose vertices are integral. Nestohedral real-rootedness conjecture. For each integral binary nestohedron $P\subseteq\mathbb R^{k+1}$, the polynomial

$$\sum_{\mathbf q\in P\cap\mathbb Z^{k+1}}N_{\mathbf q}(x)$$

has only real roots and is, therefore, log-concave.

This is a proposed nestohedral reformulation and extension of the stack-sorting descent-polynomial real-rootedness conjecture. Its real-rootedness and log-concavity remain open.