Yuan–Han conjecture on exceptional solutions of a ternary exponential equation

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. Consider the equation

$$(ak)^x+(bk)^y=((a+b)k)^z,\qquad x,y,z\in\mathbb{N}.$$

The solution $(x,y,z)=(1,1,1)$ is always present, and any solution different from it is called exceptional. Yuan–Han's conjecture. For any $k$, if $\min\{a,b\}>3$, then the equation has no exceptional solutions.

The conjecture concerns the uniqueness of the evident positive integer solution for this family of ternary purely exponential Diophantine equations. The paper proves the claim when $k>1$ and $a,b$ are both prime powers with $\min\{a,b\}>2$, so the full conjecture remains open in the stated generality.