The extremal consecutive-level conjecture for trace-Sperner families

Let $k$ and $l'$ be positive integers with $l'<k$. An $(n-l')$-trace $k$-Sperner family is a family ${\mathcal F}\subseteq 2^{[n]}$ whose trace on every $(n-l')$-set is $k$-Sperner. Extremal trace-Sperner conjecture. There exists $n_0=n_0(k,l')$ such that, if $n\ge n_0$ and ${\mathcal F}\subseteq 2^{[n]}$ is an $(n-l')$-trace $k$-Sperner family, then

$$|{\mathcal F}|\leq\sum_{i=1}^{k-l'}\binom{n}{\left\lfloor\frac{n-(k-l')}{2}+i\right\rfloor}.$$

The bound is attained by the union of the corresponding $k-l'$ consecutive levels, and the paper notes that the result is known asymptotically when $l'=1$ and $k=2$, while its main theorem establishes only the matching asymptotic upper bound for general fixed $k$ and $l'$; the exact conjecture remains open.