Uniform subexponential-growth graph hyperfiniteness

Determine whether the connected equivalence relation of every graph of uniform subexponential growth is hyperfinite.

Background

Hyperfiniteness is known for graphs of polynomial growth and for certain growth bounds below exp(rc). The question asks about the broader class of uniformly subexponential-growth graphs.

References

How about graphs of uniform subexponential growth, i.e., of growth less than $\exp(rc)$ for some $0<c<1$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 13