Is each Mader matroid a gammoid?

A Mader matroid is obtained as follows. Let G = (V,E) be an undirected graph and let S1, . . . , Sk be disjoint subsets of V . Define S := S1 ∪ · · · ∪ Sk. Let I be the collection of all subsets I of S with the property that there exist disjoint paths P1, . . . , Pt such that each Pj has its ends in different sets Si and has no internal vertices in S and such that I is a subset of the set of ends of the Pj. Then I is the collection of independent sets of a matroid, which we call a Mader matroid. The fact that it is a matroid follows from the method given in Schrijver [6], where a short proof is given of the theorem of Mader [4] on the maximum number of these paths. If k = 2, we call the Mader matroid also a Menger matroid. The matching matroids are the special case of Mader matroids where S1, . . . , Sk are all singleton subsets of V . (Matching matroids were introduced by Edmonds and Fulkerson [1], and are special cases of transversal matroids.) The question is how Mader matroids relate to known classes of matroids. The class of gammoids seems close to Mader matroids. Recall that a gammoid is obtained as follows. Let D = (V,A) be a directed graph and let S and T be disjoint subsets of V . Let I be the collection of subsets I of S such that there exist |I| disjoint directed paths from I to T . Then (S, I) is a matroid, called a gammoid.