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.