  We prove that there do not exist non-Desarguesian symplectic semifield
spreads of PG$(5,q^2)$, $q\geq 2^{14}$ even, whose associated semifield has
center containing $\mathbb{F}_q$, by proving that the only
$\mathbb{F}_q$-linear set of rank 6 disjoint from the secant variety of the
quadric Veronese variety of PG$(5,q^2)$ is a plane with three points of the
Veronese surface of PG$(5,q^6)\setminus$PG$(5,q^2)$.
