Two-part cross-intersection conjecture for two-sided 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 two-sided intersecting if any two members intersect in both parts, equivalently their intersections with $X_1$ and with $X_2$ are nonempty. Two-part cross-intersection conjecture. If ${\cal F}$ is a two-sided intersecting subfamily of ${X_1, X_2\choose k,\ell}$, then

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

The proposed extremal families combine an almost intersecting projection in one part with a pair of nonempty cross-intersecting families in the other; the source says that either this construction or its symmetric version should be largest, and gives no resolution.