Le's uniqueness conjecture for exponential Diophantine equations

Let $a$, $b$, and $c$ be coprime integers greater than $1$. Consider the Diophantine equation

$$a^x+b^y=c^z.$$

Le's uniqueness conjecture. The equation has at most one solution in integers $x,y,z>1$. This conjecture concerns the uniqueness of solutions to a class of exponential Diophantine equations. The source presents it as a conjecture arising from work on determining further solutions after one particular solution is known; its resolution status is not specified here.