  It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian
threefold in $\mathbb{P}^4 (\mathbb{F}_{q^2})$ has at most $d(q^5+q^2) + q^3 +
1$ points in common with a threefold of degree $d$ defined over
$\mathbb{F}_{q^2}$. He proved the conjecture for $d=2$. In this paper, we show
that the conjecture is true for $d = 3$ and $q \ge 7$.
