  An asymptotic formula is established for the number of rational points of
bounded anticanonical height which lie on a certain Zariski dense subset of the
biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in
$\mathbb{P}^3\times\mathbb{P}^3$. This confirms the modified Manin conjecture
for this variety, in which the removal of a thin set of rational points is
allowed.
