The maximal-element invariance conjecture for induced poset saturation

Let $\mathcal P$ be a finite poset, and let $\dot{\mathcal P}$ be the poset obtained by adding a new element greater than every element of $\mathcal P$. Let $\operatorname{sat}^*(n,\mathcal P)$ denote the induced saturation number of $\mathcal P$. Maximal-element invariance conjecture. The function $\operatorname{sat}^*(n,\mathcal P)$ is bounded if and only if $\operatorname{sat}^*(n,\dot{\mathcal P})$ is bounded. The conjecture is motivated by the fact that, for a poset without a unique maximal element, every saturated family contains the full set, so adding a maximal element may preserve boundedness of the saturation number.