The type $(4,3)$ finite asymptotic separation index matching conjecture

Let $G$ be a locally finite Borel graph with a bipartition $(A,B)$ of type $(4,3)$, meaning that vertices in $A$ have degree $4$ and vertices in $B$ have degree $3$. Let $\mathsf{asi}(G)$ denote the asymptotic separation index.

Type $(4,3)$ finite asymptotic separation index matching conjecture. If $\mathsf{asi}(G)<\infty$, then $G$ has a Borel matching covering $A$.

This is an explicitly identified special case of the finite-asymptotic-separation-index matching conjecture, and the paper states that it is already open.