Nash-Williams's matroid intersection conjecture

Let $E$ be a ground set, let $\mathfrak{F}(E)$ denote the relevant class of matroids on $E$, and for $j\in\{0,1\}$ let $\mathcal{I}_{M_j}$ be the collection of independent sets of $M_j$. A set $I$ spans a set $X$ in a matroid if $X\subseteq\operatorname{span}(I)$. Nash-Williams's matroid intersection conjecture. For every $M_0,M_1\in\mathfrak{F}(E)$, there is an $I\in\mathcal{I}_{M_0}\cap\mathcal{I}_{M_1}$ and a partition $E=E_0\sqcup E_1$ such that $I\cap E_i$ spans $E_i$ in $M_i$ for $i\in\{0,1\}$. This is an infinite-matroid analogue of matroid intersection, and the paper discusses it as an application of its main Cantor–Bernstein-type results. Its resolution is not established in the supplied text.