Equitability conjecture for matroids whose ground set is two bases

Let $M=(S,\mathcal{B})$ be a matroid. The matroid $M$ is equitable if, for every set $X\subseteq S$, there is a basis $B\in\mathcal{B}$ such that $S-B$ is also a basis and

$$\left\lfloor\frac{|X|}{2}\right\rfloor\leq |B\cap X|\leq\left\lceil\frac{|X|}{2}\right\rceil.$$

Equitability conjecture. If the ground set $S$ of $M$ can be partitioned into two bases, then $M$ is equitable. This is a simple but surprisingly open question about equitable matroids. The paper cites it as an open conjecture.