The computability conjecture for normalized phase-function integrals

Let $\mathfrak{g}$ be a simple Lie algebra with simple roots $\alpha_i$, let $V_1,V_2$ be representations of $\mathfrak{g}$ with highest weights $\lambda_1,\lambda_2$, and let $l_i$ be nonnegative integers with $l=\sum_{i=1}^n l_i$. Let $\Phi(\lambda_1,\lambda_2,\kappa)$ be the normalized phase function

$$\Phi(\lambda_1,\lambda_2,\kappa)=\prod_{j=1}^l t_j^{-(\lambda_1,\alpha_{t_j})/\kappa}(1-t_j)^{-(\lambda_2,\alpha_{t_j})/\kappa}\prod_{1\leq i<j\leq l}(t_i-t_j)^{(\alpha_{t_i},\alpha_{t_j})/\kappa}.$$

If the space of singular vectors of weight $\lambda_1+\lambda_2-\sum_{i=1}^n l_i\alpha_i$ is one-dimensional, then there is a region $\Delta=\{t\in\mathbb{R}^l\mid 0<t_{\sigma_l}<\dots<t_{\sigma_1}<1\}$ for some permutation $\sigma$, such that the integral $\int_\Delta\Phi\,dt$ can be computed explicitly and is equal to an alternating product of Euler $\Gamma$-functions up to a rational number independent of $\lambda_1,\lambda_2,\kappa$. This is verified for $\mathfrak{g}=sl(2)$ by the Selberg formula; the general assertion concerns explicit evaluations of these phase-function integrals.