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

Let $a$, $b$, and $c$ be fixed coprime positive integers with $>1$ and let $N(a,b,c)$ denote the number of solutions in positive integers $(x,y,z)$ to

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

Assume that $a$, $b$, and $c$ are not perfect powers and that $a<b$. Scott–Styer conjecture. Then $N(a,b,c)\leq 1$, except for the explicitly listed cases:

(i) $N(2,2^r-1,2^r+1)=2$, with $(x,y,z)=(1,1,1)$ and $(r+2,2,2)$, where $r\geq2$;

(ii) $N(2,3,11)=2$, with $(x,y,z)=(1,2,1)$ and $(3,1,1)$;

(iii) $N(2,3,35)=2$, with $(x,y,z)=(3,3,1)$ and $(5,1,1)$;

(iv) $N(2,3,259)=2$, with $(x,y,z)=(4,5,1)$ and $(8,1,1)$;

(v) $N(2,5,3)=2$, with $(x,y,z)=(1,2,3)$ and $(2,1,2)$;

(vi) $N(2,5,133)=2$, with $(x,y,z)=(3,3,1)$ and $(7,1,1)$;

(vii) $N(2,7,3)=2$, with $(x,y,z)=(1,1,2)$ and $(5,2,4)$;

(viii) $N(2,89,91)=2$, with $(x,y,z)=(1,1,1)$ and $(13,1,2)$;

(ix) $N(2,91,8283)=2$, with $(x,y,z)=(1,2,1)$ and $(13,1,1)$;

(x) $N(3,5,2)=3$, with $(x,y,z)=(1,1,3)$, $(1,3,7)$, and $(3,1,5)$;

(xi) $N(3,10,13)=2$, with $(x,y,z)=(1,1,1)$ and $(7,1,3)$;

(xii) $N(3,13,2)=2$, with $(x,y,z)=(1,1,4)$ and $(5,1,8)$;

(xiii) $N(3,13,2200)=2$, with $(x,y,z)=(1,3,1)$ and $(7,1,1)$.