Two-part Hilton–Milner conjecture for non-trivially intersecting families

Let $X_1$ and $X_2$ be disjoint sets with sizes $n_1$ and $n_2$, and let ${X_1, X_2\choose k,\ell}$ denote the family of sets $F$ satisfying $|F\cap X_1|=k$ and $|F\cap X_2|=\ell$. A family is intersecting if any two of its members have nonempty intersection, and non-trivially intersecting if it is intersecting but has no element common to every member. Two-part Hilton–Milner conjecture. If ${\cal F}$ is a non-trivially intersecting subfamily of ${X_1, X_2\choose k,\ell}$, then

|{\cal F}|\leq \max \left\{ \left( 1+{n_1-1\choose k-1}-{n_1-k-1\choose k-1}\right){n_2\choose \ell}, \left.{n_1\choose k}\left( 1+{n_2-1\choose \ell-1}-{n_2-\ell-1\choose \ell-1}\right) \right\}.

This conjectures that the largest non-trivially intersecting families arise from the two constructions obtained by placing the distinguished element and the exceptional set entirely in $X_1$ or entirely in $X_2$; its status is not resolved in the source.