The number of rank-k flats in a matroid with no U2,n-minor

We show that, if k and ℓ are positive integers and r is sufficiently large, then the number of rank-k flats in a rank-r matroid M with no U2,ℓ+2-minor is less than or equal to the number of rank-k flats in a rank-r projective geometry over GF(q), where q is the largest prime power not exceeding ℓ.