Katona–Lu–Milans induced forbidden subposet bound

Let $P$ be a finite poset, let $n$ be a positive integer, and let $\mathcal{F}$ be a family of subsets of an $n$-element set. Say that $\mathcal{F}$ is induced $P$-free if it contains no induced copy of $P$, and write

$$La^{\#}(n,P)=\max\{\lvert\mathcal{F}\rvert:\mathcal{F}\text{ is induced }P\text{-free}\}.$$

Katona–Lu–Milans conjecture. For every poset $P$,

$$La^{\#}(n,P)=O\left(\binom{n}{\lfloor n/2\rfloor}\right).$$

This is the induced analogue of the general extremal bound for weak forbidden subposets. The paper presents it as conjectured independently by Katona and by Lu and Milans; no resolution is supplied here.