Back-and-forth equivalent group von Neumann algebras

We prove that if $G$ and $H$ are $\alpha$-back-and-forth equivalent groups (in the sense of computable structure theory) for some ordinal $\alpha \geq \omega$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also $\alpha$-back-and-forth equivalent. In particular, if $G$ and $H$ are $\omega$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are elementarily equivalent; this is known to fail under the weaker hypothesis that $G$ and $H$ are merely elementarily equivalent. We extend this result to crossed product von Neumann algebras associated to Bernoulli actions of back-and-forth equivalent groups.