The EF1 conjecture for matroids with a common valuation

Let $M=(E,\mathcal{I})$ be an independence structure, where $E$ is a finite set and $\mathcal{I}$ is closed under taking subsets. Let $\chi(M)$ be the least number of independent sets in a partition of $E$. A matroid is an independence structure satisfying the exchange property: whenever $X,Y\in\mathcal{I}$ and $|X|>|Y|$, some $x\in X\setminus Y$ satisfies $Y\cup\{x\}\in\mathcal{I}$. For an integer $q\geq\chi(M)$, $M$ is EF1-$q$-colorable if every nonnegative additive valuation on $E$ admits an EF1 partition of $E$ into $q$ independent sets; a class has the EF1 property if every member is EF1-$q$-colorable for every such $q$. The matroid EF1 conjecture. The class of matroids satisfies the EF1 property. This is known for broad classes including base-orderable and regular matroids, but remains open for general matroids; its validity is known to follow from a conjecture of Gabow.