The level-count conjecture for forbidden subposets

Let $P$subseteq be a finite poset. Let $\operatorname{Mon}(\mathbb{Z})$ be the set of functions $f\colon\mathbb{Z}\to\{0,1\}$ such that $f(n)=1$ and $f(-n)=0$ for all sufficiently large $n$, ordered pointwise. A level is a maximal family $L\subset\operatorname{Mon}(\mathbb{Z})$ on which $\sum_n f(n)-g(n)=0$ for every $f,g\in L$. Define $l(P)$ to be the maximum number of levels whose union does not contain $P$ as a subposet. Level-count conjecture. For every finite poset $P$,

$$\operatorname{ex}(P,n)=l(P)\binom{n}{\lfloor n/2\rfloor}\bigl(1+O(1/n)\bigr).$$

This conjecture proposes a common asymptotic explanation for several known forbidden-subposet results, including those for chains, brooms, butterflies, and loops on adjacent levels of the Boolean lattice. Its general validity is not established by the surrounding discussion.