Partition regularity of the Pythagorean triples equation

Determine whether the homogeneous equation x²+y²=z² is partition regular over the positive integers, meaning that every finite coloring of the positive integers contains a monochromatic solution.

Background

The paper studies nonlinear extensions of partition regularity for Pythagorean triples. Although the equation x²+y²=z² has been verified to be 2-regular by computer-assisted work, the authors state that its partition regularity for arbitrary finite colorings remains open. Theorem 1.4 establishes 2-regularity for certain equations obtained by adding polynomial terms, but it does not resolve the unrestricted partition-regularity question for x²+y²=z².