The asymptotic Erdős–Ko–Rado conjecture for cross-t-intersecting families

Let $C(n,k,l,t)$ be the collection of pairs $(\mathcal A,\mathcal B)$ of cross-$t$-intersecting families with $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{l}$, and define

$$N(n,k,l,t)=\max_{(\mathcal A,\mathcal B)\in C(n,k,l,t)}|\mathcal A|\,|\mathcal B|.$$

The cross-$t$ Erdős–Ko–Rado conjecture. For fixed $k,l,t$, one has

$$N(n,k,l,t)=\binom{n-t}{k-t}\binom{n-t}{l-t}$$

for all $n\ge n_0(k,l,t)$. The case $t=1$ is known by the theorem of Matsumoto and Tokushige; the conjecture asserts the corresponding eventual formula for every $t$.