Nash-Williams' Matroid Intersection Conjecture

Let $M$ and $N$ be finitary matroids on the same edge set $E$. A set $I\subseteq E$ is a common independent set if it is independent in both $M$ and $N$. For a subset $A\subseteq E$, say that $I\cap A$ spans $A$ in a matroid when its closure in that matroid contains $A$.

Nash-Williams' Matroid Intersection Conjecture. The matroids $M$ and $N$ admit a common independent set $I$ for which there is a partition

$$E=E_M\sqcup E_N$$

such that $I_M:=I\cap E_M$ spans $E_M$ in $M$ and $I_N:=I\cap E_N$ spans $E_N$ in $N$.

This is a central open problem in the theory of infinite matroids and is closely related to the Packing/Covering Conjecture. The countable case was subsequently proved, but the formulation for arbitrary finitary matroids remains open.