Countable forests

Prove or disprove that ${^n}{F}=#2{^n}{F}=\chi(^n)$ for every countable forest $F$ and every dimension $n\geq2$.

Background

The main theorem establishes the equality for finite forests, while the paper proves only a more limited result for a countably infinite star.

References

Is it true that ${^{n}{F}!=!#2{^{n}{F}!=!\chi(^n)$}} for any countable forest $F$ and any $n \ge 2$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers (2512.15516 - Axenovich et al., 17 Dec 2025) in Question, Section 6.3 (Infinite graphs)