The Sperner-system strengthening of Snevily's conjecture

Let $n,s$ be integers with $n\geq 2s-1$, let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers, and let an $L$-intersecting Sperner system be a Sperner family $\mathcal{F}\subseteq 2^{[n]}$ such that $|F\cap F'|\in L$ for every distinct $F,F'\in\mathcal{F}$. The Sperner-system strengthening. Every such family satisfies

$$|\mathcal{F}|\leq\binom{n}{s}.$$

The source presents this as a stronger conjecture than Snevily's conjecture. The bound is immediate from the LYM inequality when $n<2s-1$, while the stated range is not settled in the supplied text.