The Griggs–Lu conjecture on asymptotic poset-free families

Let $P$ be a finite poset. Write

$$La(n,P):=\max\{|\mathcal F|:\mathcal F\subseteq 2^{[n]},\ \mathcal F\text{ is }P\text{-free}\}$$

and

$$La^*(n,P):=\max\{|\mathcal F|:\mathcal F\subseteq 2^{[n]},\ \mathcal F\text{ is induced }P\text{-free}\}.$$

Define $\pi(P)=\lim_{n\to\infty}La(n,P)/\binom{n}{\lfloor n/2\rfloor}$ and $\pi^*(P)=\lim_{n\to\infty}La^*(n,P)/\binom{n}{\lfloor n/2\rfloor}$ when these limits exist. For a positive integer $k$, the family $\bigcup_{i=1}^k\binom{[n]}{j+i}$ is the union of $k$ consecutive layers of $2^{[n]}$. Griggs–Lu conjecture. (i) For any poset $P$, let $e(P)$ denote the largest integer $k$ such that, for every $j$ and $n$, the family $\bigcup_{i=1}^k\binom{[n]}{j+i}$ is $P$-free. Then $\pi(P)$ exists and equals $e(P)$. (ii) For any poset $P$, let $e^*(P)$ denote the largest integer $k$ such that, for every $j$ and $n$, the family $\bigcup_{i=1}^k\binom{[n]}{j+i}$ is induced $P$-free. Then $\pi^*(P)$ exists and equals $e^*(P)$. The conjecture proposes that the asymptotic extremal sizes are obtained by taking as many consecutive middle layers as possible without creating a (induced) copy of $P$. The limits and equalities are known in many classes of posets, but remain unsettled in general.