Integral of the Vandermonde determinant.

Question

For which values of $\omega_1,\dots,\omega_d$ the following integral of the Vandermonde determinant: $$\mathcal{F}[V](\omega_1,\dots,\omega_d)=\int_\Lambda e^{-i\sum_j \theta_j \omega_j}V(e^{i\theta_1},\ldots,e^{i\theta_d})d{\theta_1}\dots d\theta_d,$$ with $V(e^{i\theta_1},\ldots,e^{i\theta_d})=\prod_{1\leq i<j\leq d}(e^{i\theta_j}-e^{i\theta_i})$ and $\Lambda=[0,2\pi]^d$, is non-negative? notice that $\mathcal{F}[V]$ is an integer function (a signed sum of Kronecker deltas).