Le's uniqueness conjecture for positive solutions of exponential equations

Let $a,b,c$ be positive integers. Le's conjecture. The equation

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

has at most one solution $(x,y,z)$ satisfying $\min(x,y,z)>1$. This conjecture is proposed after counterexamples to the Terai–Jeśmanowicz conjecture; the source does not provide a resolution.