The even-power representation conjecture for a binary quadratic form

Let $k$ be a positive integer such that $4\Vert k$ and $2k-1$ is a prime power. Write

$$k-1=ab^2,$$

where $a$ is square-free. The form $(a^2b^2,0,2k-1)$ is the binary quadratic form $a^2b^2u^2+(2k-1)v^2$. Even-power representation conjecture. Whenever this form represents a power of $k$, that power has an even exponent; equivalently, if

$$a^2b^2u^2+(2k-1)v^2=k^n,$$

then $n$ is even.