Terai's conjecture for primitive Pythagorean triples

Let $(a,b,c)$ be a primitive Pythagorean triple with

$$a^2+b^2=c^2,$$

where $a,b,c\in\mathbb{N}$, $\gcd(a,b)=1$, and $a\equiv0\pmod{2}$. Terai's conjecture. The equation

$$x^2+b^y=c^z$$

has only the solution $(x,y,z)=(a,2,2)$. The conjecture has been verified in several cases listed in the survey, but remains unresolved in general.