Stapledon decomposition conjecture for graph polynomials

Let $G$ be a graph with $d$ vertices, and let $h_G(z)$ denote its associated polynomial. A decomposition of $h_G(z)$ uses polynomials $a(z)$ and $b(z)$ satisfying the displayed symmetry conditions.

Stapledon decomposition conjecture. The polynomial $h_G(z)$ can be decomposed as

$$h_G(z)=a(z)+zb(z),$$

where

$$a(z)=z^{d}a\left(\frac{1}{z}\right),\qquad b(z)=z^{d-1}b\left(\frac{1}{z}\right),$$

and $b(z)$ and $-a(z)$ have nonnegative coefficients.

The paper presents this as a conjectural strengthening of the coefficient inequalities previously proved for $h_G(z)$. It is motivated by decomposing the contributions from the order polytopes associated with acyclic orientations of $G$; the source reports computer experimentation but gives no resolution.