On minimally k-connected matroids

A matroid M is minimally k-connected if M is k-connected and, for every e∈E(M), M\e is not k-connected. It is conjectured that every minimally k-connected matroid with at least 2(k−1) elements has a cocircuit of size k. We resolve the conjecture almost affirmatively for the case k=4 by finding the unique counterexample; and for each k⩾5, we prove that there exists a counterexample to the conjecture with 2k+1 elements.