Find all positive integers $(x,y,z,w)$ such that for any positive rational number $r=p/q$, there exist positive integers $(a,b,c,d)$ for which $$r=\frac{a^x+b^y}{c^z+d^w}.$$
For instance, for $(x,y,z,w)=(2,3,5,7)$ we can take $$(a,b,c,d)=(p^3q^7,p^5q^2,pq^3,p^2q),$$ which means that $(2,3,5,7)$ is valid.
According to this question, $(1,1,1,1)$ and $(3,3,3,3)$ are valid, while $(\text{even, even, even, even})$ is invalid.
Of course, this question is a generalization of the linked question, which is still unsolved. But any partial (positive or negative) results on other values of $(x,y,z,w)$ would also be interesting as well.
