Aharoni–Ziv's matroidal Hall conjecture for finitary matroids

Let $M$ and $N$ be finitary matroids on the common ground set $E$. For $X\subseteq E$, write $M\!\upharpoonright\!X$ for the restriction of $M$ to $X$, and $N.X$ for the contraction of $N$ to $X$. A pair $(M,N)$ is matchable when $N$ has an $M$-independent base.

Aharoni–Ziv's matroidal Hall conjecture. There is no $M$-independent base of $N$ if and only if there exists an $X\subseteq E$ such that $M\!\upharpoonright\!X$ has an $N.X$-independent base, but $N.X$ has no $M\!\upharpoonright\!X$-independent base.

This conjecture generalizes Hall's theorem and Edmonds's finite matroid intersection criterion to finitary matroids on possibly infinite ground sets. The source notes that its formulation is equivalent to the slightly stronger “unhindered” condition used by Aharoni and Ziv; no resolution is supplied here.