Keszegh–Lemons–Martin–Pálvölgyi–Patkós extension conjecture for induced saturation

Let $P$ be a poset on $[p]$, and let $\dot P$ be the poset on $[p+1]$ obtained by adjoining a new element that dominates every element of $P$. Let $\text{sat}^*(n,P)$ denote the minimum size of an induced $P$-saturated family of subsets of $[n]$. Keszegh–Lemons–Martin–Pálvölgyi–Patkós's extension conjecture.

$$\text{sat}^*(n,P)=O(1)\quad\text{if and only if}\quad\text{sat}^*(n,\dot P)=O(1).$$

This conjecture concerns which posets have bounded induced saturation number and is presented as an open research direction in the paper. No resolution is given.