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, $f(x,y)$:


<img src="http://www.tri.org.au/se/Betajointpdf.png">

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

<img src="http://www.tri.org.au/se/kthmomentofratioofBeta.png">

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).