Characterising hyperbolic hyperplanes of a non-singular quadric in $$\text {PG}(4,q)$$PG(4,q)

Let $${\mathcal {H}}$$ be a non-empty set of hyperplanes in $$\text {PG}(4,q)$$, q even, such that every point of $$\text {PG}(4,q)$$ lies in either 0, $$\frac{1}{2}q^3$$ or $$\frac{1}{2}(q^3+q^2)$$ hyperplanes of $${\mathcal {H}}$$, and every plane of $$\text {PG}(4,q)$$ lies in 0 or at least $$\frac{1}{2}q$$ hyperplanes of $${\mathcal {H}}$$. Then $${\mathcal {H}}$$ is the set of all hyperplanes which meet a given non-singular quadric Q(4, q) in a hyperbolic quadric.