The linear-or-bounded conjecture for induced poset saturation

Let $\mathcal P$ be a finite poset. For a family $\mathcal F$ of subsets of $[n]$, call $\mathcal F$ $\mathcal P$-saturated if it contains no induced copy of $\mathcal P$, but $\mathcal F\cup\{S\}$ contains an induced copy of $\mathcal P$ for every $S\notin\mathcal F$. Write $\operatorname{sat}^*(n,\mathcal P)$ for the smallest size of a $\mathcal P$-saturated family.

The linear-or-bounded conjecture. For every finite poset $\mathcal P$, the induced saturated number $\operatorname{sat}^*(n,\mathcal P)$ is either bounded, or at least linear in $n$.

Earlier results showed that the induced saturated number for every poset is either bounded or at least logarithmic in $n$, and later at least $2\sqrt n$. The conjecture asserts that the lower-growth alternative can always be strengthened to linear growth.