Quadratic Rado conjecture for homogeneous three-variable equations

Let $a,b,c\in\mathbb{N}$. A Rado triple is a triple $(a,b,c)$ such that $a=c$, $b=c$, or $a+b=c$. An equation is partition regular if every finite partition of $\mathbb{N}$ contains distinct $x,y,z$ in one cell satisfying it.

Quadratic Rado conjecture. The equation

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

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

This extends the classical linear classification of partition-regular equations to homogeneous quadratic equations. It includes the long-standing question of whether the Pythagorean equation is partition regular, and its general validity remains open.