The finite asymptotic separation index matching conjecture

Let $G$ be a locally finite Borel graph with a Borel bipartition $(A,B)$. The bipartition has combinatorial expansion if there is a constant $c>1$ such that $|N_G(S)|\geqslant c|S|$ for every finite set $S\subseteq A$. The asymptotic separation index of $G$, denoted $\mathsf{asi}(G)$, is a value in ${\mathbb N}\cup\{\infty\}$. A Borel matching covering $A$ is a Borel matching whose every vertex in $A$ is incident to an edge of the matching.

Finite asymptotic separation index matching conjecture. If the bipartition has combinatorial expansion and $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

The theorem in the paper proves this when the expansion factor is sufficiently large, and proves the stronger factor-independent bound for graphs with $\mathsf{asi}(G)\leqslant 1$. The conjecture asks whether every expansion factor greater than $1$ suffices when the asymptotic separation index is finite.