  For any fixed relatively prime positive integers $a,b$ and $c$ greater than
1, we consider the equation $a^x+b^y=c^z$ in positive integers $x,y$ and $z$.
In this note, assuming the truth of $abc$-conjecture, for any fixed $c$ we
prove that there is at most one solution to the equation, except for only
finitely many pairs of $a$ and $b$. This answers a question in the paper of
Miyazaki and Pink [T. Miyazaki and I. Pink, Number of solutions to a special
type of unit equations in two unknowns, III, arXiv:2403.20037].
