At most one solution to $a^x+b^y=c^z$ for some ranges of $a$, $b$, $c$

We consider the number of solutions in positive integers $(x,y,z)$ for the purely exponential Diophantine equation $a^x+b^y=c^z$ (with $\gcd (a,b)=1$). Apart from a list of known exceptions, a conjecture published in 2016 claims that this equation has at most one solution in positive integers $x$, $y$, and $z$. We show that this is true for some ranges of $a$, $b$, $c$, for instance, when $1<a,b<3600$ and $c<{10}^{10}$. The conjecture also holds for small pairs $(a,b)$ independent of $c$, where $2\le a,b\le 10$ with $\gcd (a,b)=1$. We show that the Pillai equation $a^x-b^y=r>0$ has at most one solution (with a known list of exceptions) when $2\le a,b\le 3600$. Finally, the primitive case of the Jeśmanowicz conjecture holds when $a\le {10}^6$ or when $b\le {10}^6$. This work highlights the power of some ideas of Miyazaki and Pink and the usefulness of a theorem by Scott.