Rota's Basis Conjecture for Matroids with Density Close to One

Rota’s basis conjecture (RBC) states that given a collection B of n bases in a matroid M of rank n, one can always find n disjoint rainbow bases with respect to B. We show that if M is a matroid having n + k elements, then one can construct n - k^3 disjoint rainbow bases, where b is a constant depending only on k.