Conjecture on consecutive-layer extremal families avoiding a fixed poset

Let $P$ be a fixed finite poset, let $e(P)$ be the largest integer $k$ such that the union of the $k$ middle layers of the Boolean lattice is $P$-free, and let $\operatorname{La}^c(n,P)$ be the maximum size of a $P$-free family contained in $e(P)+1$ consecutive layers. If the limit exists, define

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

Consecutive-layer conjecture. For any fixed poset $P$,

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

This is presented as a weaker version of the general extremal-poset conjecture. The source highlights the diamond poset as an important unresolved case of the stronger conjecture.