Hi! Thank you for your thoughts! In the problem description, it now says $c$ instead of $d$, as it should be. So $c$ is just some given constant complex vector. I also have a feeling a closed form does not exist. However, you brought up an interesting question. Indeed, I am not really after a closed form here. Instead, I am interested in finding the $\tau$, $1\leq\tau\leq N$, that maximizes that expression for a given vector $c$. However, since I believe that one can get any $tau$ by changing $c$, I feel I must have a closed form expression to solve this maximization.

That's a great derivation! Thanks a lot!

Thanks for your comment! I also noticed numerically that it does not seem to converge. I agree that if it converges, it should converge to a scaled identity. However, I found it strange that it doesn't converge, since it is related to the inverse of a Wishart matrix (where the eigenvalues of the 4 submatrices in the Wishart matrix are fixed).