Characterising hyperbolic hyperplanes of a non-singular quadric in PG (4, q)

Let H be a non-empty set of hyperplanes in PG(4,q), q even, such that every point of PG(4,q) lies in either 0, 1/2q³ or 1/2(q³+q²²) hyperplanes of H, and every plane of PG(4,q) lies in 0 or at least 1/2q hyperplanes of H. Then H is the set of all hyperplanes which meet a given non-singular quadric Q(4, q) in a hyperbolic quadric.S.G. Barwick, Alice M.W. Hui, Wen-Ai Jackson, Jeroen Schillewaer