Nonnegativity conjecture for the summed edge-deletion gamma-polynomial

Let $G=(V,E)$ be a 2-connected graph. For each edge $ij\in E$, set $c^{ij}(t):=h^*_{P_G}(t)-h^*_{P_{G\setminus ij}}(t)$, and let $\gamma(c^{ij}(t))$ denote the $\gamma$-polynomial associated with this palindromic difference. Define

$$Z_G(t):=\sum_{ij\in E}\gamma(c^{ij}(t))=\sum_{ij\in E}\gamma\bigl(h^*_{P_G}(t)-h^*_{P_{G\setminus ij}}(t)\bigr)=\sum_{ij\in E}\bigl(\gamma_{P_G}(t)-\gamma_{P_{G\setminus ij}}(t)\bigr).$$

Summed edge-deletion nonnegativity conjecture. The polynomial $Z_G(t)$ has nonnegative coefficients. This is proposed as a stronger statement whose gamma-nonnegativity would imply the existence of an edge with gamma-nonnegative deletion difference; the supplied text gives no resolution status.