The cross-intersection product criterion for uniform families

Let $n,k,l$ be positive integers with $k+l<n<\max\{2k,2l\}$. For each $j\geq 0$, let

$$\mathcal A_j^{(k)}:=\{A\in\binom{[n]}k:1\in A\}\sqcup\{A\in\binom{[n]}k:A\cap[j+2]=[j+2]\setminus\{1\}\},$$

and

$$\mathcal B_j^{(l)}:=\{B\in\binom{[n]}l:1\in B\}\setminus\{B\in\binom{[n]}l:B\cap[j+2]=\{1\}\}.$$

Let $M(n,k,l)$ be the maximum of $|\mathcal A||\mathcal B|$ over cross-intersecting families $\mathcal A\subset\binom{[n]}k$ and $\mathcal B\subset\binom{[n]}l$. The cross-intersection product criterion. If $|\mathcal A_j^{(k)}||\mathcal B_j^{(l)}|<\binom{n-1}{k-1}\binom{n-1}{l-1}$ for all $j\geq0$, then

$$M(n,k,l)=\binom{n-1}{k-1}\binom{n-1}{l-1}.$$

Moreover, every pair of cross-intersecting families attaining this maximum is a pair of common stars: there is some $i\in[n]$ such that $\mathcal A=\{A\in\binom{[n]}k:i\in A\}$ and $\mathcal B=\{B\in\binom{[n]}l:i\in B\}$. The criterion addresses the unresolved range $k+l<n<\max\{2k,2l\}$, extending the known star-product result beyond the cases $n\geq\max\{2k,2l\}$; its truth is intended to characterize when stars are optimal and unique.