A proof for the conjecture of characteristic function of the generalized skew-elliptical distributions

In the paper of Genton and Loperfido (Genton and Loperfido (2005)), the authors introduced the multivariate generalized skew-elliptical distributions, which is a family of skewed distributions that contains the more familiar skew-normal and skew-Student-t distributions. In the same paper the authors conjectured the structure of the characteristic function of the proposed family of distributions. In this short letter we prove their conjecture.