Canonical Ramsey status of the regular hexagon

Determine whether the regular hexagon is canonically Ramsey.

Background

The paper proves that regular polygons with a prime number of sides are canonically Ramsey, but explains that its proof does not extend to composite-sided regular polygons. For every composite r at least 6, the paper gives colourings of powers of the regular r-gon that avoid monochromatic and rainbow scaled copies within those powers. This obstruction does not settle whether some other ambient configuration can witness canonical Ramseyness for the regular hexagon, motivating the explicit question posed in the concluding remarks.