Normalized average flat-size conjecture for complex-representable matroids

Let $M$ be a simple complex-representable matroid, and let $r(F)$ denote the rank of a flat $F$ of $M$. Consider the average, over all flats $F$ of $M$, of the quantity

$$\frac{|F|}{\binom{r(F)+1}{2}}.$$

Normalized average flat-size conjecture. This average is at most $1+o_n(1)$. The source presents this as a strengthening of the preceding asymptotic flat-size conjecture; the meaning of $o_n(1)$ is asymptotic in the number $n$ of elements.