Ordinal-sum factorization conjecture for polar preorder-polytope h-star polynomials

Let $\tau$ be the ordinal sum of preorders $\tau_1$ and $\tau_2$, and let $\mathcal R_\tau^\vee$ denote the polar dual of the translated reflexive preorder polytope. Ordinal-sum factorization conjecture.

$$h^*(\mathcal R_\tau^\vee,t)=h^*(\mathcal R_{\tau_1}^\vee,t)h^*(\mathcal R_{\tau_2}^\vee,t).$$

The source states this as a conjectural factorization for ordinal sums and gives no resolution.