Eggs in PG(4n−1,q), q even, containing a pseudo-pointed conic

An ovoid of PG(3,q) can be defined as a set of q2+1 points with the property that every three points span a plane and at every point there is a unique tangent plane. In 2000 M.R. Brown (J. Geom. 67 (2000) 61) proved that if an ovoid of PG(3,q), q even, contains a pointed conic, then either q=4 and the ovoid is an elliptic quadric, or q=8 and the ovoid is a Tits ovoid. Generalising the definition of an ovoid to a set of (n−1)-spaces of PG(4n−1,q), J.A. Thas (Rend. Mat. (6) 4 (1971) 459) introduced the notion of pseudo-ovoids or eggs: a set of q2n+1(n−1)-spaces in PG(4n−1,q), with the property that any three egg elements span a (3n−1)-space and at every egg element there is a unique tangent (3n−1)-space. We prove that an egg in PG(4n−1,q), q even, contains a pseudo-pointed conic, that is, a pseudo-oval arising from a pointed conic of PG(2,qn), q even, if and only if the egg is elementary and the ovoid is either an elliptic quadric in PG(3,4) or a Tits ovoid in PG(3,8).