Extension conjecture for planes in the Klein quadric

Let $q$ be a prime power, let $\mathcal{Q}=Q^+(5,q)$ be the Klein quadric, and fix a plane $\pi$ in $\mathcal{Q}$. Let $\mathcal{P}$ be the set of planes in $\mathcal{Q}$ disjoint from $\pi$; the supplied context states that $\mathcal{P}$ is a spanning $q$-divisible set of $q^3$ planes in $\operatorname{PG}(5,q)$, with every pair of planes intersecting in a point. Extension conjecture. The extension problem for $\mathcal{P}$ admits a solution for all prime powers $q$. Such a solution would provide the missing extension to a tight irreducible affine vector space partition of $\operatorname{PG}(6,q)$ of type $4^{q^3}$; the supplied text does not establish existence of the extension.