Assume they are four independent beta random variables $X_i$, with means $\mu_i$.

Note that the density functions would depend on the observed samples.

We could then test the hypothesis that $\mu_1\mu_2>\mu_3\mu_4$ by setting up a quadruple integral of the joint density function over the set

$\{ (x_1,x_2,x_3,x_4)\mid x_1x_2>x_3x_4 \}$.

If this integral is small, we reject the hypothesis $\mu_1\mu_2>\mu_3\mu_4$.