Scott–Styer's conjecture on solutions to a ternary purely exponential Diophantine equation

Let $\mathbb{N}$ be the set of positive integers, and let $a$, $b$, and $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$, assumed not to be perfect powers. Let $N(a,b,c)$ denote the number of solutions $(x,y,z)\in\mathbb{N}^3$ to

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

Scott–Styer's conjecture. If $a<b$, then $N(a,b,c)\leq 1$, apart from the thirteen explicitly listed exceptional cases and solutions in the source statement: the family $N(2,2^r-1,2^r+1)=2$ for $r\geq2$, and the cases $(2,3,11)$, $(2,3,35)$, $(2,3,259)$, $(2,5,3)$, $(2,5,133)$, $(2,7,3)$, $(2,89,91)$, $(2,91,8283)$, $(3,5,2)$, $(3,10,13)$, $(3,13,2)$, and $(3,13,2200)$ with the displayed solution tuples. This conjecture seeks a complete finite description of when the equation has more than one solution; several upper-bound results are known, but the general assertion remains open.