On a conjecture concerning the number of solutions to $a^x+b^y=c^z$

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min \{a,b,c\}>1$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x+b^y=c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then, taking $a<b$, we must have $a=2$, $(b,c)\equiv (1,17)$, $(13,5)$, $(13,17)$, or $(23,17)24$, and $(a,b,c)$ must satisfy further strong restrictions, including $c>{10}^{14}$. These results support a conjecture of the last two authors.