Asymptotic bound for average plane-size in rank-4 complex-representable matroids

Let $M$ be a simple $n$-element, rank-$4$, complex-representable matroid that is not the union of two lines. Its average plane-size is the average of $|P|$ over the rank-$3$ flats $P$ of $M$. Average plane-size conjecture. The average plane-size in $M$ is at most $6+o_n(1)$. The Dowling geometries described in the source have average plane-size tending to $6$, motivating the bound; the conjecture concerns the asymptotically sharp behavior in rank $4$.