Conjecture on the number of solutions to $a^x+b^y=c^z$

Let $a$, $b$, and $c$ be integers greater than one with $\gcd(a,b)=1$, and consider solutions in positive integers $(x,y,z)$ to

$$a^x+b^y=c^z.$$

One-solution conjecture. There is at most one solution $(x,y,z)$, except when $(a,b,c)$ or $(b,a,c)$ is one of $(5,2,3)$, $(7,2,3)$, $(3,2,11)$, $(3,2,35)$, $(3,2,259)$, $(3,4,259)$, $(3,16,259)$, $(5,2,133)$, $(3,10,13)$, $(89,2,91)$, $(91,2,8283)$, $(3,5,2)$, $(3,13,2)$, $(3,13,4)$, $(3,13,16)$, $(3,13,2200)$, or $(2^n-1,2,2^n+1)$ for a positive integer $n\geq 2$. The conjecture proposes a complete list of triples for which more than one positive-integer solution can occur. The stated result is presented as a conjecture in the paper, and no resolution is supplied in the given context.