Induced forbidden subposet density conjecture

Let $P$ be a poset. Write $B_n$ for the Boolean poset of all subsets of $[n]$, and let $La^*(n,P)$ be the maximum size of an induced $P$-free subposet of $B_n$. Define

$$e^*(P)=\text{the largest integer }k\text{ such that, for every }j\text{ and }n,\ \bigcup_{i=1}^k\binom{[n]}{j+i}\text{ is induced }P\text{-free}.$$

Induced forbidden subposet density conjecture. The limit

$$\pi^*(P)=\lim_{n\to\infty}\frac{La^*(n,P)}{\binom{n}{\lfloor n/2\rfloor}}$$

exists and

$$\pi^*(P)=e^*(P).$$

This is the induced analogue of the ordinary forbidden subposet density conjecture. The source notes that only limited results were known: an asymptotic upper bound was established for every poset, and the induced analogue was proved for tree posets with an $o(1)$ error term, but the conjecture remains open in general.