Fermat–Catalan's conjecture for generalized Fermat equations

Let $p,q,r$ be positive integers, and call a solution $(\alpha,\beta,\gamma)$ of $x^p+y^q=z^r$ primitive when $\gcd(\alpha,\beta,\gamma)=1$. Fermat–Catalan's conjecture. For any triple of exponents $(p,q,r)$ satisfying

$$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1,$$

the set of primitive solutions $(\alpha,\beta,\gamma)$ of

$$x^p+y^q=z^r$$

is finite. This is a finiteness conjecture for primitive solutions of generalized Fermat equations and is presented in the source as closely related to Beal's conjecture. Its status is not specified in the supplied text.