A note on a conjecture of Jeśmanowicz

Let a, b, c be relatively prime positive integers such that ${\displaystyle a^2+b^2=c^2}$. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${\displaystyle {\left(an\right)}^x+{\left(bn\right)}^y={\left(cn\right)}^z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${\displaystyle {(u^2-v^2)}^x+{\left(2uv\right)}^y={(u^2+v^2)}^z}$, for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.