Jesmanowicz's conjecture on exponential Diophantine equations

Let $a,b,c$ be relatively prime positive integers satisfying

$$a^{2}+b^{2}=c^{2}.$$

A solution $(x,y,z)$ in positive integers is nontrivial if it is not $(2,2,2)$. Jesmanowicz's conjecture. The equation

$$a^{x}+b^{y}=c^{z}$$

has no nontrivial solutions in positive integers $x,y,z$; equivalently, its only solution is the trivial solution $(x,y,z)=(2,2,2)$. The conjecture extends the classical characterization of primitive Pythagorean triples and is known in several infinite and parametrized families, but the general statement remains open.