Fixed non-bipartite graphs attaining the color-grouping bound

Determine whether there exist an integer $m\geq2$ and an $(m+1)$-chromatic graph $H$ such that $#2{^n}{H}=\lceil\chi(^n)/m\rceil$ for every dimension $n\geq2$.

Background

Grouping a proper coloring into classes of at most mm colors gives a general upper bound; the unresolved issue is whether one fixed non-bipartite graph can attain that bound in every dimension.

References

Is there $m \ge 2$ and an $(m+1)$-chromatic graph $H$ such that $#2{n}{H}= \lceil \chi(n)/m \rceil$ for all $n\ge 2$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question, Section 6.2 (Growing dimension)