On partitioning two matroids into common independent subsets

Let M1 and M2 be two matroids on the same ground set S. We conjecture that if there do not exist disjoint subsets A1,A2,…,Ak+1 of S, such that ⋂ispM1(Ai)≠Ø, and similarly for M2, then S is partitioned into k sets, each independent in both M1 and M2. This is a possible generalization of König's edge-coloring theorem. We prove the conjecture for the case k=2 and for a regular case, in which both matroids have the same rank d, and S consists of k·d elements. Finally, we prove another special case related to a conjecture of Rota.