Finiteness conjecture for minor-minimal matroids not coverable by independent sets

Let $M$ be a matroid. A collection of independent sets covers $M$ if every element of $M$ belongs to at least one set in the collection. Finiteness conjecture. For every positive integer $k$, there are finitely many minor-minimal simple matroids that cannot be covered by $k$ independent sets. The case $k=2$ is known: the ten minor-minimal simple matroids are explicitly determined, but the assertion for all positive integers remains open.