The conjecture for $x^2-2=y^n$

Let $x,y,n\in\mathbb{N}$ satisfy $\gcd(x,y)=1$ and $n>2$. The conjecture for $x^2-2=y^n$. The equation

$$x^2-2=y^n$$

has no solutions $(x,y,n)$. The source presents this as an unresolved conjecture and notes that only upper bounds are known.