Expectation of the pseudoinverse of a complex Gaussian matrix with non identically distributed columns

Question

Let us define the $M \times N$ matrix $\boldsymbol{C}=\left[\boldsymbol{c}_1 \cdots \boldsymbol{c}_N\right]$, where $\boldsymbol{c}_n \sim \mathcal{CN}\left(\boldsymbol{0}, \boldsymbol{R}_n\right)$ (i.e., the $n$th column of matrix $\boldsymbol{C}$ is a zero-mean circularly symmetric complex Gaussian random vector with correlation matrix $\boldsymbol{R}_n$). Do you know how to calculate the expectation of the pseudoinverse of $\boldsymbol{C}$? I have conducted numerous simulations using Matlab, and it appears that this expectation should be equal to $\boldsymbol{0}$. However, I am uncertain whether this is accurate or how to formally prove it.