Generalized Fermat conjecture on primitive solutions

Let $p,q$ and $r$ be positive integers satisfying

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

A solution $(X,Y,Z)$ is called primitive here when $XYZ\ne0$ and $\gcd(X,Y)=1$. Generalized Fermat conjecture. All primitive solutions of

$$X^p+Y^q=Z^r$$

come from the following ten identities: $1^p+2^3=3^2$, $7^2+2^5=3^4$, $13^2+7^3=2^9$, $17^3+2^7=71^2$, $11^4+3^5=122^2$, $1549034^2+33^8=15613^3$, $96222^3+43^8=30042907^2$, $2213459^2+1414^3=65^7$, $15312283^2+9262^3=113^7$, and $76271^3+17^7=21063928^2$. The conjecture is a proposed classification of all such solutions when the reciprocal-exponent sum is less than one; its resolution status is not specified in the supplied text.