Terai's conjecture for the generalized Ramanujan equation

Let $k>1$ be an integer, and let $x,y,z\in\mathbb{N}$. Consider the Diophantine equation

$$x^2+(2k-1)^y=k^z.$$

Terai's conjecture. The equation has only the solution

$$(x,y,z)=(k-1,1,2).$$

The conjecture concerns a generalized Ramanujan–Nagell equation. The paper proves it when $4\Vert k$, $2k-1$ is a prime power, and $4\leq k\leq 1000$; the general case remains open.