Suppose $\Omega \in \mathbb{R}^{m \times n}$ has i.i.d. $\mathcal{N}(0,1)$ entries and $\Sigma = \text{diag}(\lambda_1,\dots,\lambda_n)$. Here $m < n$ so $\Omega$ has full row rank with probability $1$. I would like to understand the distribution of the matrix $\Omega \Sigma^2 \Omega^T$. Specifically, is this matrix Wishart? I am ultimately interested in calculating $\mathbb{E}_{\Omega}\text{trace}((\Omega \Sigma^2 \Omega^T)^{-1})$. If $\Omega \Sigma^2 \Omega^T$ is Wishart, I believe I can use properties about the inverse Wishart distribution to calculate this trace.
What I currently have is the following: if $\Omega_{i}$ denotes the $i$-th column of $\Omega$, then $$\Omega \Sigma^2 \Omega^T = \sum_{i=1}^n \lambda_i^2 \Omega_i \Omega_i^T$$ so $\mathbb{E}_{\Omega}\Omega\Sigma^2\Omega^T = \text{trace}(\Sigma^2)I_m$ since each $\Omega_i$ is isotropic. However, the elements in this sum are not identically distributed so I think it may not be Wishart. Is there a related distribution that this may follow?