The unboundedness conjecture for glued posets

Let $\mathcal P_1$ and $\mathcal P_2$ be finite posets, and let $\mathcal P_2*\mathcal P_1$ denote their gluing, obtained by placing every element of $\mathcal P_1$ below every element of $\mathcal P_2$. Let $\operatorname{sat}^*(n,\mathcal P)$ denote the induced saturation number. Glued-poset unboundedness conjecture. The function $\operatorname{sat}^*(n,\mathcal P_2*\mathcal P_1)$ is unbounded if and only if at least one of $\mathcal P_1$ and $\mathcal P_2$ has unbounded saturation number. This would extend the paper's established result in the case where at least one glued poset has the unique cover twin property.