Let $$ X \sim \mathcal{W}_{q} (n, \Sigma) \; \; n > q$$ and $$ Y \sim \mathcal{W}^{-1}_{q} (n, \Sigma^{-1}) \; \; n>q$$
Where $\mathcal{W}$ denotes the Wishart distribution and $\mathcal{W}^{-1}$, the inverse Wishart distribution. $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive definite).
What is the distribution of $XY$ if $X$ and $Y$ are not independent?