The boundedness conjecture for induced saturation under adjoining a maximum

Let $P$ be a poset, and let $\dot P$ denote the poset obtained from $P$ by adjoining a largest element. Write ${\rm sat}^*(n,P)$ for the minimum size of a maximal induced $P$-free subposet of the Boolean lattice $B_n$.

Adjoining-a-maximum conjecture. The quantity ${\rm sat}^*(n,P)$ is bounded as a function of $n$ if and only if ${\rm sat}^*(n,\dot P)$ is bounded as a function of $n$.

The observation preceding the conjecture shows one implication at the level of saturating families when $P$ has no largest element, but the authors could verify neither direction in general. The equivalence is known only in special cases and remains open.