Generating weakly 4-connected matroids

We prove that, if M is a weakly 4-connected matroid with |E(M)|⩾7 and neither M nor M∗ is isomorphic to the cycle matroid of a ladder, then M has a proper minor M′ such that M′ is weakly 4-connected and |E(M′)|⩾|E(M)|−2 unless M is some 12-element matroid with a special structure.