Dominance conjecture for descent polynomials of spinal graphs

Let $G$ be a spinal graph, meaning a graph with a Hamiltonian path. Let $\mathcal{A}^{\downarrow}_G$ and $\mathcal{A}^{\uparrow}_G$ be the families of lower and upper $G$-cyclic orders, let $\mathcal{F}_G$ be the flow polytope, and define

$$P_{\mathcal{A},\mathsf{des}}(t)=\sum_{\gamma\in\mathcal{A}}t^{\mathsf{des}(\pi(\gamma))}.$$

Here $\lhd$ denotes dominance order on the coefficients.

Dominance conjecture.

$$P_{\mathcal{A}^{\downarrow}_G,\mathsf{des}}(t)\lhd h^*_{\mathcal{F}_G}(t)\lhd P_{\mathcal{A}^{\uparrow}_G,\mathsf{des}}(t).$$

The conjecture relates descent enumerators of upper and lower cyclic orders to the $h^*$-polynomial of a flow polytope. The source presents it as a related conjecture but gives no resolution evidence.