Erdős–Graham conjecture for equal-colored variables

An equation is partition regular if every finite coloring of $\mathbb{N}$ admits a solution whose variables all have the same color. Erdős–Graham conjecture. For every finite coloring of $\mathbb{N}$, there exist $x,y$ of the same color such that $x^2+y^2$ or $x^2-y^2$ is a perfect square. The two-color case for $x^2+y^2$ is known, but the stated finite-coloring problem remains open, including the weaker requirement that only $x$ and $y$ have the same color.