Fair representation in dimatroids

For a simplicial complex C denote by β(C) the minimal number of edges from C needed to cover the ground set. If C is a matroid then for every partition A1,…,Am of the ground set there exists a set S∈C meeting each Ai in at least |Ai|β(C) elements. We conjecture a slightly weaker statement for the intersection of two matroids: if D=P∩Q, where P, Q are matroids on the same ground set V, and β(P), β(Q)≤k, then for every partition A1,…,Am of the ground set there exists a set S∈D meeting each Ai in at least 1k|Ai|−1 elements. We prove that when m=2 there is a set meeting each Ai in at least (1k−1|V|)|Ai|−1 elements.