Real-rootedness and root-location conjecture for generalized snake posets

Let $\mathbf{w}$ be a generalized snake word of length $n+1$. Let $h^*(\mathbf{w};z)$ denote its $h^*$-polynomial, and let $L(\mathcal{O}(P(\mathbf{w}));t)$ denote the Ehrhart polynomial of the order polytope $\mathcal{O}(P(\mathbf{w}))$. Conjecture for generalized snake words. All roots of $h^*(\mathbf{w};z)$ are real and negative, and all roots of $L(\mathcal{O}(P(\mathbf{w}));t)$ are contained in the disk

$$\left|z-\frac{n+4}{2}\right|\leq\frac{n+2}{2},\qquad z\in\mathbb{C},$$

with axis of symmetry $x=\frac{-n-4}{2}$. The conjecture was verified for snake words of length up to $9$, but the general assertions remain open.