On Millsʹs conjecture on matroids with many common bases

In this paper, we shall prove a conjecture of Mills: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n1 circuit-hyperplanes and from the other by relaxing n2 circuit-hyperplanes, where n1 and n2 are non-negative integers such that n1+n2⩽k