The Rado triple conjecture for homogeneous quadratic equations

Let $a,b,c\in\mathbb{N}$. An equation is partition regular if every finite coloring of $\mathbb{N}$ admits a monochromatic solution. A triple $(a,b,c)$ is a Rado triple in the sense used for homogeneous quadratic equations.

Rado triple conjecture. The equation

$$ax^2+by^2=cz^2$$

is partition regular if and only if $(a,b,c)$ is a Rado triple.

The necessity follows from Rado's theorem together with the implication that partition regularity of $P(x^2,y^2,z^2)=0$ implies partition regularity of $P(x,y,z)=0$. No instance of a Rado triple is currently known for which the required partition regularity has been proved; notable test cases include $x^2+y^2=z^2$, $x^2+2y^2=z^2$, and $x^2+y^2=2z^2$.