A chain theorem for 4-connected matroids

A matroid M is said to be k-connected up to separators of size l if whenever A is (k-1)-separating in M, then either |A|⩽l or |E(M)-A|⩽l. We use si(M) and co(M) to denote the simplification and cosimplification of the matroid M. We prove that if a 3-connected matroid M is 4-connected up to separators of size 5, then there is an element x of M such that either co(M⧹x) or si(M/x) is 3-connected and 4-connected up to separators of size 5, and has a cardinality of |E(M)|-1 or |E(M)|-2.