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@Suvrit: Just a clarification, are you sure that $\phi(X)$ is strictly convex on $X>0$? For instance, assume that $X^*>0$ is a trace-1 minimum of $\phi$, then for every traceless $X_0$ s.t. $X^*+X_0> 0$, and $\sum_i v_i^\top X_0 v_i=0$, $X^*_0:=X^*+X_0>0$ is also a minimum of $\phi$.
@Suvrit: Great comments, thanks! As you pointed out, the main problem is that, even if $\nabla \phi(X)=0$ admits a (unique) solution, since the Picard iteration lives in the set of trace-1 positive semi-definite matrices it can tend towards a singular matrix. (Btw, I used Brouwer’s theorem w.r.t. the aforementioned compact set). My (vague) idea is to exploit the additional fact that the $\varepsilon I$-perturbed Picard iteration always converges, and then use some kind of "exchanging limits law’’ between $\varepsilon\to 0$ and $k\to \infty$. However, to prove the last step is the main issue...