Borel incomparability of isomorphism and quasi-isomorphism for p-local groups

Let $m,n\geq3$ and let $p,q$ be primes. Write $\oiso_{m,p}$ for isomorphism equivalence on the space of $p$-local torsion-free abelian groups of rank $m$, and $\oqiso_{n,q}$ for quasi-isomorphism on the space of $q$-local torsion-free abelian groups of rank $n$. For equivalence relations $E$ and $F$, write $E\perp_B F$ when they are Borel incomparable. Isomorphism/quasi-isomorphism incomparability conjecture.

$$\oiso_{m,p}\perp_B\oqiso_{n,q}.$$

This asks whether the isomorphism/quasi-isomorphism distinction is sufficient to yield Borel incomparability even when the ranks, and hence the dimensions of the acting groups, differ. The surrounding results establish incomparability for isomorphism versus isomorphism and quasi-isomorphism versus quasi-isomorphism with distinct locality primes, while the mixed case remains open.