Two-character sets, Veronese varieties and Grassmannian of lines

We show that, for any q, the pointset covered by the conic planes of a quadric Veronesean of \(\textrm{PG}(3,q)\) $\text{PG}(3,q)$ is a two character set with respect to hyperplanes of \(\textrm{PG}(9,q)\) $\text{PG}(9,q)$ .