Bukh–Griggs–Lu conjecture for the extremal number of a poset

Let $P$ be a finite poset and let $\mathcal{B}_n=2^{[n]}$ be the Boolean lattice. Write $La(n,P)$ for the largest size of a $P$-free subfamily of $\mathcal{B}_n$. Define $e(P)$ to be the largest integer $\ell$ such that, for every $j$ and $n$,

$$\bigcup_{i=1}^{\ell}\binom{[n]}{i+j}$$

is $P$-free. Bukh–Griggs–Lu conjecture.

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

This conjecture proposes the asymptotic size of the largest $P$-free family in the Boolean lattice. The paper presents it as a conjecture of Griggs and Lu and independently Bukh; no resolution is supplied in the provided text.