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 2 k + 1 elements.