LOCAL complexity of the Lovász Local Lemma on subexponential-growth graphs

Determine whether the LOCAL complexity of the distributed Lovász Local Lemma on every bounded-degree subexponential-growth graph class F is Θ(log*(n)), equivalently whether randomized O(log(n)) complexity can always be deterministically reduced to O(log*(n)).

Background

This is presented as a special case of the Chang–Pettie conjecture. The question concerns a general speed-up from randomized to deterministic distributed computation on subexponential-growth graph classes.

References

Is the LOCAL complexity of the distributed Lov\'{a}sz Local Lemma on ${\bf F}$ equal to $\Theta(\log*(n))$? That is, is there a speed-up result from randomized LOCAL complexity $O(\log(n))$ to deterministic LOCAL complexity $O(\log*(n))$ for ${\bf F}$?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem 9, Section 7 (LCL problems)

Let ${\bf F}$ be a class of graphs of bounded degree and of subexponential growth. Is the LOCAL complexity of the distributed Lov\'asz Local Lemma on ${\bf F}$ equal to $\Theta(\log*(n))$? That is, is there a speed-up result from randomized LOCAL complexity $O(\log(n))$ to deterministic LOCAL complexity $O(\log*(n))$ for ${\bf F}$?

— From descriptive to distributed  (2502.15347 - Grebík et al., 21 Feb 2025) in Problem 9, Section 11 (Open problems), LCL problems