The original orchard-planting problem asks for the maximum number of $3$-point lines attainable by a configuration of points in the plane. I am interested in its natural generalization for (three-dimensional) space, that is in the maximum number of $4$-point planes attainable by a configuration of points in the space, no $3$ of which are collinear and no $5$ of which are coplanar. I guess this should be a known problem, but, surprisingly, I googled nothing about it.
More precisely, I am interested only in an upper bound of this number and shall be quite happy if it is $o(n^2)$. In fact, if it is $c n^2$ for $c<1/12$ then we already obtain an improvement over a straightforward asymptotic lower bound for the minimal number $\rho(n)$ of planes in space needed to a straight-line crossing free drawing of the complete graph $K_n$ in Theorem 12 from our paper "Drawing Graphs on Few Lines and Few Planes". For instance, $\rho(6)=4$, that is for $K_6$ we need exactly $4$ planes.
PS. This approach to the lower bound for our problem failed, because for it are essential only planes containing fours of points in a non-convex position, but it is too specific to ask for a reference.