Palindromic decomposition conjecture for symmetric edge-polytope edge deletion

Let $G$ be a 2-connected graph and let $ij$ be any edge of $G$. Let $\Gamma$ be a unimodular triangulation of the subcomplex of $P_{G\setminus ij}$ consisting of all facets visible by $e_{ij}$. If the distinct lattice distances from $e_{ij}$ to facets of $\Gamma$ are $h_1>h_2>\cdots>h_k$, define

$$\Gamma_s:=\langle\sigma\text{ facet of }\Gamma\mid h_\sigma\ge h_s\rangle$$

for $s=1,\ldots,k$, and set $h_{\Gamma_0}(t):=0$. Palindromic decomposition conjecture.

$$h^*_{P_G}(t)-h^*_{P_{G\setminus ij}}(t)=2t\sum_{s=1}^{k}\left(\left(h_{\Gamma_s}(t)-h_{\Gamma_{s-1}}(t)\right)\sum_{r=0}^{h_s-1}t^r\right).$$

Moreover, for every $s=1,\ldots,k$, the polynomial

$$\left(h_{\Gamma_s}(t)-h_{\Gamma_{s-1}}(t)\right)\sum_{r=0}^{h_s-1}t^r$$

is palindromic, has nonnegative coefficients, and has the same center as the left-hand side. This proposed formula would decompose the edge-deletion difference into palindromic pieces with nonnegative coefficients; the supplied text gives no resolution status.