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OP wrote:

what is known about the moments of this ratio?

I have not seen the paper ... but one does not even need to derive the distribution of the ratio in order to derive the moments of the ratio. In particular:

If $X$ ~ $Beta(a,b)$ and $Y$ ~ $Beta(c,d)$ are independent, then the joint pdf of $(X,Y)$ is, say, $f(x,y)$:

Then, the $k$-th raw moment of the ratio $\frac{X}{Y}$ can be derived immediately as:

where I am using the Expect function from the mathStatica add-on to Mathematica to automate the nitty-gritties for me (I am one of the developers of the former).