Frankl–Wang product Hilton–Milner conjecture for cross-intersecting families

Let $n$, $k$, and $\ell$ satisfy

$$n \ge 2k > 2\ell \ge 4.$$

Let $\mathcal{F} \subset \binom{[n]}{k}$ and $\mathcal{G} \subset \binom{[n]}{\ell}$ be non-trivial cross-intersecting families, meaning that $F\cap G\ne\emptyset$ for every $F\in\mathcal{F}$ and $G\in\mathcal{G}$. For $X\in\binom{[2,n]}{k}$ and $Y\in\binom{[2,n]}{\ell}$ with $X\cap Y\ne\emptyset$, define

$$M_k(n,k,\ell)=\binom{n-1}{k-1}-\binom{n-\ell-1}{k-1}+1,$$

and

$$M_\ell(n,k,\ell)=\binom{n-1}{\ell-1}-\binom{n-k-1}{\ell-1}+1.$$

Frankl–Wang conjecture. Under these hypotheses,

$$|\mathcal{F}||\mathcal{G}|\le M_k(n,k,\ell)M_\ell(n,k,\ell).$$

The right-hand side is attained by the natural Hilton–Milner-type pair $\mathcal{H}_k(X,Y)$ and $\mathcal{H}_\ell(Y,X)$. The conjecture is a product analogue of the Hilton–Milner theorem for unequal uniformities; it was disproved by a robust family of counterexamples.