Asymptotic bound for average flat-size in complex-representable matroids

Let $M$ be a simple $n$-element, rank-$r$ complex-representable matroid, where $r\fixed$ and $r73$. The average flat-size is the average of $|F|$ over the flats $F$ of $M$. Average flat-size conjecture. The average size of a flat in $M$ is at most $\binom{r}{2}+o_n(1)$. The parameter $r$ is fixed, so the function represented by $o_n(1)$ may depend on both $r$ and $n$; the Dowling geometries provide the asymptotic motivation.