Generalized Pythagorean partition-regularity conjecture

Let $a,b,c\in\mathbb{N}$, and call the equation $ax^2+by^2=cz^2$ partition regular when every finite colouring of $\mathbb{N}$ admits a solution $(x,y,z)\in\mathbb{N}^3$ whose three entries have the same colour.

Generalized Pythagorean partition-regularity conjecture. The equation

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

is partition regular if and only if $c\in\{a,b,a+b\}$.

Rado's theorem gives the necessity of the condition $c\in\{a,b,a+b\}$, but its sufficiency remains open. Indeed, it is unknown whether any choice of $a,b,c\in\mathbb{N}$ makes this equation partition regular. The paper proves a special case when $a,b,c$ are perfect squares and either $a=c$ or $a+b=c$.