Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), i.e., $X | V=v \sim \mathrm{Unif}(\mu-\sigma \sqrt{v},\mu+\sigma \sqrt{v})$ and latent variable $V\sim \Gamma(3/2,1/2)$, as shown in Qin et al. "Scale Mixture Models with Applications to Bayesian Inference".

Question: is it possible to generalize to the multivariate normal distribution as a mixture of uniform distributions over high dimensional balls parameterized by some latent variable?