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Arguably, the most important open problem in the theory of $q$-analogs ofdesigns is the question for the existence of a $q$-analog $D$ of the Fanoplane. 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 mustalways exist at least one non-$\alpha$-point. For the binary case $q = 2$, OlofHeden and Papa Sissokho have improved this result in 2016 by showing that thenon-$\alpha$-points must form a blocking set with respect to the hyperplanes.  In this article, we show that a hyperplane consisting only of $\alpha$-pointsimplies the existence of a partiton of the symplectic generalized quadrangle$W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho isgeneralized to all primes $q$ and all even values of $q$.

On $\alpha$-points of $q$-analogs of the Fano plane