Equicovering matroids by distinct bases

We ask, and answer, the following question: when is it possible to cover the elements of a given matroid M by p distinct bases so that each element is covered exactly k times? Our result is weaker than the Cyclic Order Conjecture of Wiedemann [9] (and of Kajitani, Ueno and Miyano [7]), but stronger than similar previous results; it could represent a first step towards resolving the conjecture.