Prove that $\min_{\pi \in \mathbb{R}^n} \operatorname{tr}\left[(X^T \operatorname{diag}(\pi)\: X)^{-1}\right]$ is convex opt problem

Question

Prove that the following is convex optimization problem: \begin{align*} \min_{\pi \in \mathbb{R}^n} & \quad \operatorname{tr}\left[(X^T \operatorname{diag}(\pi)\: X)^{-1}\right] \\ \text{s. t.} & \quad \pi \geq 0, \\ & \quad \mathbf{1}^T \pi = 1 \end{align*} $X \in \mathbb{R}^{n \times p}$ and $\pi \in \mathbb{R}^n$

Hint: Treat the trace as a sum of $p$ variables, where each of the $p$ variables is set to some expression involving $X$ and $\pi$. Transform the equation defining each of these elements into an SDP constraint, using the Schur lemma:

So I wrote the whole matrix $X^T \operatorname{diag}(\pi) X$ as a sum of variables but what I can I do with the inverse. Any hints?