Vertex degree-Ramsey boundedness of square grids

Determine whether the family of square grid graphs P_n \square P_n is vertex-R_\Delta-bounded; equivalently, determine whether their vertex degree-Ramsey numbers are bounded by a function depending only on the grid maximum degree and the number of colors.

Background

The paper identifies square grids as a specific unresolved case of the general vertex degree-Ramsey problem. The n by n square grid is defined as the Cartesian product P_n \square P_n, and every such grid has bounded maximum degree.

Although the paper establishes vertex-R_\Delta-boundedness for trees, closed blowups of trees, and bounded-treewidth graphs, it does not determine whether the family of square grids has the same property. The authors explicitly present this question at the end of the paper.