# Equitability of matroids

Is it true that if the ground set S of a matroid M can be partitioned into 2 bases, then for any set $X \subseteq S$ there is a basis B such that $S \setminus B$ is also a basis and $\lfloor|X|/2 \rfloor \leq |B\cap X| \leq \lceil|X|/2 \rceil$?
The matroids with the above property are called equitable. The conjecture would be a consequence of the conjecture on Cyclic orderings of matroids: if there is a cyclic ordering such that any $|S|/2$ consecutive elements form a basis, then it is easy to see that for any $X \subseteq S$ one of these cyclically consecutive bases has the above property.