On nontrivial cross-2-intersecting families

Two families $\mathcal{A}\subseteq\binom{[n]}{k}$ and $\mathcal{B}\subseteq\binom{[n]}{\ell}$ are said to be nontrivial cross-$t$-intersecting if $|A \cap B| \geq t$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$, and $|\bigcap_{A\in \mathcal{A}\cup \mathcal{B}}A|<t$. In this paper, we determine the upper bound on $|\mathcal{A}||\mathcal{B}|$ of two nontrivial cross-$2$-intersecting families $\mathcal{A}\subseteq\binom{[n]}{k}$ and $\mathcal{B}\subseteq\binom{[n]}{\ell}$ for any positive integers $n,k,\ell$ with $k\geq \ell \geq 3$ and $n \geq 3(k-1)$. Moreover, we characterize the extremal families attaining this bound. This settles the last unsolved case of a recent result by He, Li, Wu and Zhang (J. Combin. Theory Ser. A, 217 (2026) 106095).