Aharoni-Berger conjecture

Is it true that if $$M_1, \ldots, M_k$$ are matroids on the same ground set S and $$\sum_{i=1}^k r_i (X_i) \geq (k-1)l$$ for all partitions $$X_1, \ldots, X_k$$ of S, then there exists a common independent set of size $$l$$ ?

Remarks

Ryser's conjecture asks whether for any k-uniform k-partite hypergraph $$\tau \leq (k-1)\nu$$ holds, where $$\tau$$ is the minimum size of a vertex cover and $$\nu$$ is the maximum number of disjoint hyperedges. The case k=2 is just Kőnig's theorem, while the case k=3 was proved by Aharoni . Just as Edmonds' matroid intersection theorem generalizes Kőnig's theorem, the Ryser conjecture is generalized by the Aharoni-Berger conjecture. It was proved for k=3 by Aharoni and Berger .