Ehrhart $h^*$-polynomials of $(132,213)$-avoiding permutation polytopes: A repair cone and eventual real-rootedness

Let $P_d(132,213)$ be the convex hull of the permutations in $S_d$ that avoid $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial, defined by $\sum_{m\ge0}|mP_d(132,213)\cap\mathbb{Z}^d|t^m=\frac{H_d(t)}{(1-t)^d}$. We give an explicit lattice equivalence between this polytope, a path-L