Planar grid Ramsey property

Prove or disprove that every 2-coloring of the Euclidean plane contains a monochromatic unit-copy of $P_n\square P_n$ for every positive integer $n$.

Background

The question strengthens the planar C_4 problem and is motivated by the known unavoidable monochromatic 3-term arithmetic progression result.

References

Is it true that $2 \xrightarrow{2} P_n\square P_n$ for every $n \in $?

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