Minimum convex-region area ruling out six-colorings

Determine the minimum area of a convex region for which, under the conditions of Theorem 2—proper locally finite map-type coloring, no trichromatic vertex of degree greater than 3, and no boundary arcs of unit curvature—a proper 6-coloring cannot exist.

Background

The paper proves that seven colors are required for coloring the entire plane under specified map-type conditions, and it notes that the same reasoning remains valid inside a circle of radius 3. The first concluding question asks how far this finite-region conclusion can be localized and seeks the smallest convex region whose area alone guarantees the nonexistence of a 6-coloring under the hypotheses of Theorem 2.