Existence of nonsymmetric eggs in odd characteristic

Determine whether an egg O(n, m, q) exists for odd q when m is not equal to 2n.

Background

The paper studies generalized ovoids, also called eggs, in PG(2n + 2m − 1, q), subject to dimensional and tangent-space conditions. Known parameter restrictions imply n < m ≤ 2n and force m = 2n when q is even. The existence of eggs with m different from 2n in odd characteristic remains unresolved.