Partition regularity conjecture for generalized Pythagorean pairs

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

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

An equation is partition regular with respect to $x,y$ if every finite coloring admits a solution in which $x$ and $y$ have the same color. A non-zero integer is a square if it is the square of an integer.

Generalized Pythagorean-pair conjecture. If at least one of $ac$, $bc$, or $(a+b)c$ is a square, then $ax^2+by^2=cz^2$ is partition regular with respect to $x,y$.

This is presented as a variant of the main Rado triple conjecture covering partition regularity of pairs. The source does not report a resolution, so the assertion remains open.