Hegedüs' conjecture for cross-intersecting projective subspaces over fields of order other than two

Prove Hegedüs' conjecture for cross-intersecting pairs of families of projective subspaces in an n-dimensional projective space over a finite field of order q when q is not equal to 2; namely, establish the upper bound m ≤ 2^{n+1} − 2 for the number m of pairs.

Background

The paper studies cross-intersecting pairs of families of projective subspaces, where each paired intersection A_i ∩ B_i is empty and every cross-intersection A_i ∩ B_j with i < j is nonempty. Hegedüs conjectured that the number of such pairs satisfies m ≤ 2^{n+1} − 2. The paper proves the corresponding upper bound when q = 2, but leaves the conjecture unresolved for fields with q ≠ 2.