An application of $abc$-conjecture to a conjecture of Scott and Styer on purely exponential equations

For any fixed relatively prime positive integers $a,b$ and c greater than 1, we consider the equation aˣ+bʸ=cᶻ 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].