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,…,A_m of the ground set there exists a set S∈C meeting each A_i in at least [Formula presented] 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,…,A_m of the ground set there exists a set S∈D meeting each A_i in at least [Formula presented]|A_i|−1 elements. We prove that when m=2 there is a set meeting each A_i in at least ([Formula presented]−[Formula presented])|A_i|−1 elements.