Miyazaki's shuffle conjecture for Terai–Jeśmanowicz equations

Let $a,b,c,p,q,r$ be fixed positive integers satisfying

$$a^p+b^q=c^r,\qquad \min\{p,q,r\}>1.$$

Assume further that $a<b$, $(a,b,c)\ne(2,7,3)$, and $(a,b,c)\ne(2,2^k-1,2^k+1)$ for a positive integer $k$. Consider positive integer solutions $(X,Y,Z)$ of

$$c^X+b^Y=a^Z.$$

Miyazaki's shuffle conjecture. If $q=r=2$ and $b+1=c$, the only solution is $(X,Y,Z)=(1,1,p)$; otherwise, there are no solutions. The source presents this as a final open conjectural variant of the Terai–Jeśmanowicz problem.