The asymptotic extremal conjecture for forbidden subposets

Let $P$ be a finite poset. For a family $\mathcal F\subseteq 2^{[n]}$, let $La(n,P)$ be the largest size of a $P$-free family, and let $La^*(n,P)$ be the largest size of an induced $P$-free family. Let $e(P)$ and $e^*(P)$ be the maximum integers defined by the corresponding unions of consecutive levels being, respectively, $P$-free and induced $P$-free. The asymptotic extremal conjecture.

$$La(n,P)=(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor},\qquad La^*(n,P)=(e^*(P)+o(1))\binom{n}{\lfloor n/2\rfloor}.$$

These conjectured asymptotics generalize the theorem of Erdős for chains and are fundamental problems in forbidden subposet theory. The source states that the conjecture has been verified for several classes of posets but remains open in general.