The FKM conjecture on pairwise partition regularity of quadratic equations

Let $a,b,c\in \mathbb{Z}$ be nonzero, and consider the homogeneous quadratic equation

$$a x^2+b y^2=c z^2.$$

An equation is partition regular with respect to $x,y$ if every finite coloring of the positive integers admits a monochromatic choice of $x,y$ for which the equation has a solution in the remaining variable $z$.

FKM's pairwise partition-regularity conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $a x^2+b y^2=c z^2$ is partition regular with respect to $x,y$.

The assumptions are motivated by necessary conditions developed in the paper, and the conjecture is known when $ac$ or $bc$ is a square. The remaining cases, including the equation $x^2+y^2=2z^2$ with respect to any two variables, remain open.