The six-exception conjecture for prime-base exponential Diophantine equations

Let $\mathbb{P}$ be the set of positive rational primes, and let $S(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of

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

where $a,b,c\in\mathbb{P}$, $a<b$, and the three primes are distinct. The six-exception conjecture. We have $S(a,b,c)\leq 1$, except for the following cases: (i) $S(2,3,5)=2$, with $(x,y,z)=(1,1,1)$ and $(4,2,2)$; (ii) $S(2,3,11)=2$, with $(x,y,z)=(1,2,1)$ and $(3,1,1)$; (iii) $S(2,5,3)=2$, with $(x,y,z)=(1,2,3)$ and $(2,1,2)$; (iv) $S(2,7,3)=2$, with $(x,y,z)=(1,1,2)$ and $(5,2,4)$; (v) $S(3,5,2)=3$, with $(x,y,z)=(1,1,3)$, $(1,3,7)$, and $(3,1,5)$; (vi) $S(3,13,2)=2$, with $(x,y,z)=(1,1,4)$ and $(5,1,8)$. This conjecture has been restated and proven, so it is now a theorem.