The small-parameter trace-Sperner asymptotic conjecture

Let $f(n,k,l)$ be the maximum size of an $l$-trace $k$-Sperner family ${\mathcal F}\subseteq 2^{[n]}$. For a family, being $l$-trace $k$-Sperner means that the trace on every $l$-set is $k$-Sperner. Small-parameter trace-Sperner conjecture. For every pair of integers $k\leq l'$, one has

$$f(n,k,n-l')=\Theta_{k,l'}\left(\frac{1}{n^{l'-k+1}}\binom{n}{\lfloor n/2\rfloor}\right).$$

The conjecture is proposed as a common generalization of the preceding results and conjectures for the case $k\leq l'$. The paper records partial estimates, including the cases $k=1$ and uniform families, but does not establish the asserted order of magnitude in general.