Siar–Keskin conjecture for the Pell and Pell Lucas families

Let $P_k$ and $Q_k$ denote the Pell and Pell Lucas numbers, respectively, and let $n,x$ be natural numbers. For an odd integer $k>3$, consider the exponential Diophantine equation

$$(2^n-2)((2P_k)^n-2)=x^2.$$

Siar–Keskin's Pell-family conjecture. For every odd integer $k>3$, the equation has only the solution

$$(n,x)=(2,Q_k).$$

The authors state that they are unable to prove this conjecture. The case $k=3$ is excluded because the corresponding equation has an additional known solution.