Uniformly cross intersecting families

Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff |A∩B|=ℓ for all A∈ A and B∈B. Denote by P _e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P _e (n) is Θ(2 ⁿ ), by Frankl and R?dl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with |A||B| = $ left( {{*{20}c} {2ell } ell } right) $ left( {begin{array}{*{20}c} {2ell } ell end{array} } right) 2 ^n−2ℓ = Θ(2 ⁿ /$ sqrt ell $ sqrt ell ), and conjectured that this is best possible. Consequently, Sgall asked whether or not P _e (n) decreases with ℓ.