Assume a random matrix, denoted as X, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{i,j})$, In this case, the matrix $G= XX^{T}$ follows Wishart distribution and its general inverse $G^{-1}$ follows inverse non-singular Wishart  distribution with entries following zero mean and variance $G_{ij} \sim \frac{1}{\sigma_{i,j}*T(T-n-1))}$ from this paper .
I am curious about a case where the entries of $X$, $X_{ij}\sim \mathcal{N}(\mu_{i,j},\sigma_{i,j})$. What would be the distribution of the general inverse $G^{-1}$ for such non-zero-centered data matrix?