Decomposition of oriented k-partition-connected digraphs

Let D=(V,A) be a digraph whose underlying graph is k-partition-connected, and let $$r_0 \in V$$ be a node of in-degree 0. Suppose that the in-degree of every other node is at least k. Is it true that D can be decomposed into k weakly connected spanning subgraphs, so that every node $$v \in V-r_0$$ has positive in-degree in each subgraph?

Remarks

Let $$M_1$$ be the cycle matroid of the underlying undirected graph of D, and let $$M_2$$ be the partition matroid where each class consists of the set of edges entering a given node $$v \in V-r_0$$ . The problem can be formulated as follows:

Is it true that if both $$M_1$$ and $$M_2$$ contain k disjoint spanning sets, then there are k disjoint common spanning sets?

This statement is known to be true for strongly base orderable matroids ; however, the cycle matroid of a graph is not necessarily base orderable.