The complementary exceptional propus-parameter conjecture

Let $s$ be an odd prime with $s\equiv1\pmod{4}$. Consider the Diophantine equation

$$\alpha^2+\beta^2=s(\alpha-\beta).$$

Complementary exceptional propus-parameter conjecture. In positive integers, this equation has a unique solution $(a,b)$ satisfying $1<a\le(s-1)/2$; moreover, $a-b$ is either a square or twice a square. This conjecture concerns the additional normalized propus parameter set for $v=s^2$ and remains open in general.