The basis graph of a bicolored matroid

Let ϕϕ be a 2-coloring of the elements of a matroid MM. The bicolor basis graph   of MM is the graph G(B(M),ϕ)G(B(M),ϕ) with vertex set given by the set of bases of MM in which two bases BB and B′B′ are adjacent if B′=(B−e)∪fB′=(B−e)∪f for some elements e∈Be∈B and f∈B′f∈B′ with ϕ(e)≠ϕ(f)ϕ(e)≠ϕ(f). Let MM be a matroid with at least one circuit, we prove that G(B(M),ϕ)G(B(M),ϕ) is connected for every 2-coloring ϕϕ of MM if and only if MM is a connected matroid.