  Arguably, the most important open problem in the theory of $q$-analogs of
designs is the question for the existence of a $q$-analog $D$ of the Fano
plane. It is undecided for every single prime power value $q \geq 2$.
  A point $P$ is called an $\alpha$-point of $D$ if the derived design of $D$
in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must
always exist at least one non-$\alpha$-point. For the binary case $q = 2$, Olof
Heden and Papa Sissokho have improved this result in 2016 by showing that the
non-$\alpha$-points must form a blocking set with respect to the hyperplanes.
  In this article, we show that a hyperplane consisting only of $\alpha$-points
implies the existence of a partiton of the symplectic generalized quadrangle
$W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is
generalized to all primes $q$ and all even values of $q$.
