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, $f(x,y)$:
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:
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). If desired, one can express the solution slightly more neatly as:
$$\frac{B(a+k,b) B(c-k,d)}{B(a,b) B(c,d)}$$
where $B$ denote the Euler beta function.