Unavoidable parallel minors of regular matroids

We prove that, for each positive integer k , every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M ∗ ( K 3 , k ) , M ( W k ) , M ( K k ) , the cycle matroid of the graph obtained from K 2 , k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K 3 , k by adding a complete graph on the vertex class with three vertices.