On partitioning two matroids into common independent subsets

Let image and image be two matroids on the same ground set S. We conjecture that if there do not exist disjoint subsets image of S, such that image, and similarly for image, then S is partitioned into k sets, each independent in both image and image. This is a possible generalization of Königʹs edge-coloring theorem. We prove the conjecture for the case image and for a regular case, in which both matroids have the same rank d, and S consists of image elements. Finally, we prove another special case related to a conjecture of Rota.