Keskin–Siar–Karaatli conjecture for a quadratic Diophantine equation

Let $n$ be a nonnegative integer, and let $k$ be an integer. Consider the Diophantine equation

$$x^{2}-kxy+y^{2}-2^{n}=0.$$

A positive integer solution means $x,y>0$, and a positive odd integer solution means $x,y>0$ are both odd. Keskin–Siar–Karaatli conjecture. (i) If $n$ is odd and $n>2$, then for $k>2^{n}-2$ the equation has no positive integer solution. If $k\leq 2^{n}-2$ and the equation has a solution, then $k$ is even. (ii) If $n$ is even and $k>2^{n}-2$, then the equation has no positive odd integer solution. If $k\leq 2^{n}-2$ and the equation has a positive odd integer solution, then $k$ is even. The paper proves this conjecture, resolving the previously open classification problem for the equation.