Theta-graph h-star polynomial conjecture

Let $P_k$, $P_\ell$, and $P_m$ be paths of lengths $k$, $\ell$, and $m$, respectively, and let $G$ be the graph obtained by identifying the three starting nodes and the three ending nodes. Theta-graph h-star polynomial conjecture. The $h^*$-polynomial of the cosmological polytope of $G$ is

$$h^*(\mathbb{C}_G;z)=(1+3z)^{k+\ell+m}-(2z)^{k+\ell}(1+3z)^m-(2z)^{k+m}(1+3z)^\ell-(2z)^{\ell+m}(1+3z)^k-(2z)^{k+\ell+m}+3\cdot(2z)^{k+\ell+m}.$$

This is a conjectural formula for a family of $2$-connected graphs motivated by computational evidence. The family includes $K_{2,3}$, and specializing the formula to that graph recovers the normalized-volume formula attributed in the paper to Landin; a general proof is not supplied.