An application of $p$-adic Baker method to a special case of Jeśmanowicz' conjecture

In 1956, Jeśmanowicz conjectured that, for any positive integer $n$, the Diophantine equation ${\left((f^2-g^2)n\right)}^x+{\left((2fg)n\right)}^y={\left((f^2+g^2)n\right)}^z$ has only the positive integral solution $(x,y,z)=(2,2,2)$, where $f$ and $g$ are positive integers with $f>g$, gcd $(f,g)=1$, and $f\not\equiv g(\mathrm{mod}2)$. Let $r=6k+2$, $k\in \mathbb{N}$, $k\ge 25$. In this paper, combining $p$-adic form of Baker method with some detailed computation, we prove that if $n$ satisfies $n\equiv 0,6,9(\mathrm{mod}12)$, $f=g+1$ and $g=2^r-1$, then the conjecture is true.