Keszegh–Lemons–Martin–Pálvölgyi–Patkós dichotomy conjecture for induced poset saturation

Let $P$ be a finite poset, and let $\mathrm{sat}^{\star}([2]^n,P)$ denote its induced saturation function in the Boolean lattice. Keszegh–Lemons–Martin–Pálvölgyi–Patkós' dichotomy conjecture. There exists a constant $C_P$ such that either

$$\mathrm{sat}^{\star}([2]^n,P)\leq C_P$$

for all $n$, or

$$\mathrm{sat}^{\star}([2]^n,P)\geq n+1$$

for all sufficiently large $n$. This conjecture sharpens the known bounded-versus-unbounded dichotomy for induced poset saturation; its status is unclear from the supplied text.