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

An ovoid of $\PG(3,q)$ can be defined as a set of $q^2+1$ 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 (\cite{BROWN2000b}) proved that if an ovoid of $\PG(3,q)$, $q$ even, contains a conic, then the ovoid is an elliptic quadric. Generalising the definition of an ovoid to a set of $(n-1)$-spaces of $\PG(4n-1,q)$ J. A. Thas \cite{THAS1971} introduced the notion of pseudo-ovoids or eggs: a set of $q^{2n}+1$ $(n-1)$-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-conic, that is, a pseudo-oval arising from a conic of $\PG(2,q^n)$, if and only if the egg is classical, that is, arising from an elliptic quadric in $\PG(3,q^n)$.