On a conjecture on exponential Diophantine equations

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, $b\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.