Let $G$ be a standard $d \times d$ Wishart random matrix and consider the problem of maximizing the function $$ f(M) = \mathbb{E}\Big[\mathrm{tr}((G + M^{-1})^{-1})\Big], $$ over the class of real symmetric positive definite $M$ that have unit trace. It is possible to show that this is maximized with the choice $M = (1/d) I_d$ where $d$ denotes the dimension. (This follows by noting that the map $M \mapsto (G + M^{-1})^{-1}$ is operator concave.)
My question is how this generalizes to the following problem. Let $P$ denote a positive definite matrix and consider $$ f_P(M) = \mathbb{E}\Big[\mathrm{tr}((G + P M^{-1}P)^{-1})\Big]. $$ Is it possible to determine $M^\star_P$ such that $$ f_P(M^\star_P) = \sup_{M \succ 0, \mathrm{tr}(M) = 1} f_P(M)? $$ The previous case reduces to $M^\star_{I_d} = \tfrac{1}{d} I_d$. A possible conjecture is that something like $$ M^\star_P = \tfrac{1}{\mathrm{tr}(P^{-2})} P^{-2} $$ holds, but I am unable to prove it.
