  In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on
the intersection of a surface of degree $d$ and a non-degenerate Hermitian
surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in
the case when $d=2$. In this paper, we prove that the conjecture is true for
$d=3$ and $q \ge 8$. We further determine the second highest number of rational
points on the intersection of a cubic surface and a non-degenerate Hermitian
surface. Finally, we classify all the cubic surfaces that admit the highest and
second highest number of points in common with a non-degenerate Hermitian
surface. This classifications disproves one of the conjectures proposed by
Edoukou, Ling and Xing.
