On preserving matroid 3-connectivity relative to a fixed basis

We show that for any 3-connected matroid M on a ground set of at least four elements such that M does not contain any 4-element fans, and any basis B of M, there exists a set K⊆E(M) of four distinct elements such that for all k∈K, si(M/k) is 3-connected whenever k∈B, and co(M∖k) is 3-connected whenever k∈E(M)−B. Moreover, we show that if no other elements of E(M)−K satisfy this property, then M necessarily has path-width 3.