Snevily's conjecture for non-uniform L-intersecting families

Let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers and $K=\{k_{1},k_{2},\ldots,k_{r}\}$ be a set of $r$ positive integers with

$$\max\{\ell_i\}<\min\{k_j\}.$$

Let an $L$-intersecting family be a family $\mathcal{F}$ such that $|F\cap F'|\in L$ for every distinct $F,F'\in\mathcal{F}$. Snevily's conjecture. If

$$\mathcal{F}\subseteq\bigcup_{i=1}^{r}\binom{[n]}{k_i},$$

then

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

This conjecture generalizes the Ray-Chaudhuri–Wilson theorem, which proves the claim when $|K|=1$. Polynomial-method results establish asymptotically matching bounds and Snevily proved special cases, but the general conjecture is not resolved in the supplied text.