The boundedness conjecture for gluing posets

Let $\mathcal P_1$ and $\mathcal P_2$ be finite posets with bounded saturation number, 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. Gluing-boundedness conjecture. The glued poset $\mathcal P_2*\mathcal P_1$ also has bounded saturation number. This is a special case of the preceding unboundedness conjecture and is presented as a desired extension of the paper's boundedness theorem to arbitrary posets.