Mixed-uniformity product Hilton–Milner conjecture

Let $n\geq 2k>2\ell\geq 4$. Fix sets $F_0\in\binom{[2,n]}{k}$ and $G_0\in\binom{[2,n]}{\ell}$ with $F_0\cap G_0\neq\emptyset$, and define

$$\mathcal{F}_0=\{F_0\}\cup\{F\in\binom{[n]}{k}:1\in F,\ F\cap G_0\neq\emptyset\},$$ $$\mathcal{G}_0=\{G_0\}\cup\{G\in\binom{[n]}{\ell}:1\in G,\ G\cap F_0\neq\emptyset\}.$$

Two families are cross-intersecting if every member of one intersects every member of the other, and a family is non-trivial if the intersection of all its members is empty. Mixed-uniformity product Hilton–Milner conjecture. Suppose that $\mathcal{F}\subset\binom{[n]}{k}$ and $\mathcal{G}\subset\binom{[n]}{\ell}$ are non-trivial cross-intersecting families. Then

$$|\mathcal{F}||\mathcal{G}|\leq |\mathcal{F}_0||\mathcal{G}_0|.$$

The conjecture asks for the extremal product when the two cross-intersecting families have different uniformities, with the displayed pair as the proposed extremal construction. The supplied text does not state a resolution, so the conjecture remains open.