Upper bound conjecture for the h-star polynomial of cosmological polytopes

Let $G=(V,E)$ be a graph, and let $h^*(\mathbb{C}_G;z)$ denote the $h^*$-polynomial of its cosmological polytope. For polynomials with nonnegative coefficients, write $f\preceq g$ when $f$ is coefficientwise bounded above by $g$. Upper bound conjecture. For every graph $G=(V,E)$,

$$h^*(\mathbb{C}_G;z)\preceq (1+3z)^{|E|}.$$

This would give the conjectured tight upper bound $4^{|E|}$ for the normalized volume, complementing the exponential lower bound established earlier in the paper. The claim is presented as a conjecture motivated by the expected complexity of computing the associated wavefunction, and no resolution is supplied here.