Terai–Jeśmanowicz' conjecture for coprime bases

Let $a,b,c,p,q,r\in\mathbb{N}$ satisfy

$$a^{p}+b^{q}=c^{r},$$

with $a,b,c,p,q,r\geq 2$ and $\gcd(a,b)=1$. Terai–Jeśmanowicz' conjecture. The equation

$$a^{x}+b^{y}=c^{z}$$

has only the solution $(x,y,z)=(p,q,r)$ with $x,y,z>1$. The paper states that this conjecture is false and describes infinitely many counterexamples.