The Aharoni–Berger–Kotlar–Ziv fair representation conjecture

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be matroids on the same ground set $E$, and assume that $E$ can be partitioned into $k$ independent sets in each matroid. For a set $X\subseteq E$, say that $X$ represents a set $S\subseteq E$ almost $\alpha$-fairly if

$$|S\cap X|\ge \lfloor\alpha|S|\rfloor-1.$$

It represents a partition almost $\alpha$-fairly if it does so for every part. Aharoni–Berger–Kotlar–Ziv conjecture. For every partition of $E$, there exists a common independent set $X$ of $\mathcal{M}_1$ and $\mathcal{M}_2$ that represents the partition almost $1/k$-fairly. This conjecture seeks a uniformly fair common independent representative under simultaneous matroid independence constraints.