The distribution shown is not a Wishart distribution, it is a multivariate t-distribution. This is the marginal likelihood of $X$ after integrating the covariance matrix out of the Gaussian sampling model.

The distribution shown is not a Wishart distribution, it is a multivariate t-distribution. This is the marginal distribution of $X$ after integrating the covariance matrix out of the Gaussian sampling model. The expression may not be correct (I haven't checked carefully, but it doesn't look quite right). A good reference for the derivation is section 1.4.3 of the book Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference by Gamerman and Lopes.