Pseudo-ovals on elliptic quadrics

Determine whether every pseudo-oval on a non-singular elliptic quadric E(5, q) in PG(5, q), for odd q, is a pseudo-conic.

Background

For odd q, pseudo-ovals in PG(5, q) may lie on non-singular elliptic quadrics. The paper notes that such objects correspond to point sets on a Hermitian variety with a tangent-plane avoidance property, while pseudo-conics are known to arise as intersections of suitable hyperbolic and elliptic quadrics. Whether all pseudo-ovals on E(5, q) are of this classical type remains open.