Full-level extremality conjecture for posets of unequal heights

Let $l(P)$ denote the height of a poset $P$, namely the length of its longest chain. For posets $P,Q$ with $l(P)>l(Q)$, let $c(Q,\mathcal F_n)$ be the number of copies of $Q$ in a family $\mathcal F_n\subseteq 2^{[n]}$. Unequal-height full-level conjecture. There exists a sequence of $P$-free families $\mathcal F_n\subseteq 2^{[n]}$, each a union of full levels, such that

$$La(n,P,Q)=(1+o(1))c(Q,\mathcal F_n).$$

This is proposed as a salvage of the broader full-level principle after the source gives counterexamples when no height condition is imposed. The source does not report a proof or a disproof of this restricted assertion.