Miyazaki's shuffle conjecture for Jeśmanowicz equations

Let $f,g$ be positive integers with $f>g$, $\gcd(f,g)=1$, and $fg\equiv0\pmod 2$. Consider positive integer solutions $(x,y,z)$ of

$$(f^2+g^2)^x+(2fg)^y=(f^2-g^2)^z.$$

Miyazaki's shuffle conjecture. If $f=g+1$, the only solution is $(x,y,z)=(1,1,2)$; otherwise, there are no solutions. The source describes this as unsolved, with verification in several special cases.