Characteristic restrictions for nonclassical good pseudo-ovoids

Determine whether an odd-characteristic good pseudo-ovoid in PG(4n − 1, q) that is neither classical nor of Kantor–Knuth type necessarily has q equal to a power of 3.

Background

The paper classifies good pseudo-ovoids in odd characteristic into several cases, including classical, Kantor–Knuth, and a further Veronese-surface case. Known examples outside the first two types occur for q = 3^h, and the authors ask whether this observed characteristic restriction is necessary.