Two-coloring of the plane and monochromatic four-cycles

Determine whether every 2-coloring of the Euclidean plane contains a monochromatic unit-copy of the four-cycle C_4.

Background

The paper proves that a monochromatic unit-copy of C_4 is unavoidable in every 2-coloring of the plane if the planar chromatic number is 7, but this conditional conclusion is not known unconditionally.

References

Is it true that $2 \xrightarrow{2} C_4$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question 1, Section 6.1 (Planar case)