The dichotomy conjecture for induced poset saturation

Let $P$ be a poset, let $B_n$ be the Boolean lattice on $[n]$, and let ${\rm sat}^*(n,P)$ denote the minimum size of a maximal induced $P$-free subposet of $B_n$.

Induced saturation dichotomy conjecture. Either there exists a constant $K_P$ such that

$${\rm sat}^*(n,P)\le K_P,$$

or, for every $n$,

$${\rm sat}^*(n,P)\ge n+1.$$

This strengthens the paper's dichotomy theorem, which gives the weaker alternative lower bound ${\rm sat}^*(n,P)\ge\log_2 n$ in the unbounded case. The proposed linear lower bound remains open.