The diamond-free Lubell-function extremal conjecture

For $n\ge 4$, let $\mathcal{D}_2$ be the diamond poset, let $\mathcal{F}\subset 2^{[n]}$ be a $\mathcal{D}_2$-free family, and let $\overline{\mathcal{F}}=\{[n]\setminus F\mid F\in\mathcal{F}\}$ be its conjugate. Let $C_i(S,T)$, for $i=1,2,3$, denote the three specified construction families from the paper, and write $\bar{h}_n(\mathcal{F})$ for the normalized Lubell-type quantity used there.

Diamond-free Lubell-function conjecture. Every such family satisfies

$$\bar{h}_n(\mathcal{F})\le 2+\frac{\lfloor n^2/4\rfloor}{n(n-1)},$$

and equality holds if and only if, up to relabelling the elements of $[n]$, either $\mathcal{F}$ or $\overline{\mathcal{F}}$ is $C_i(S,T)$ for some $i\in\{1,2,3\}$ with $\bigl||S|-|T|\bigr|\le 1$.

The conjecture is intended to identify the exact extremal configurations controlling the asymptotics of diamond-free families and would imply the suspected limiting value for the sequence discussed in the paper. Because the excerpt does not define $\bar{h}_n$ or the constructions $C_i(S,T)$ in full, those definitions should be checked against the source; the parser supplies no resolution status.