Let $U$ be some convex subset of $\mathbb{C}^d$, and define the set of square matrices $\mathcal{U} \subset \mathbb{C}^{d \times d}$ to be the set of all convex combinations of $x y^*$ such that $x \in U, y \in U$. (Here $^*$ denotes the conjugate transpose.)

Is it always true that if a matrix belongs to $\mathcal{U}$ and is Hermitian and positive semidefinite, then it can be written as a convex combination of terms $x x^*$ with $x \in U$?

**Edit**: The conjecture was shown to be false in the most general case in the answer below, but is it possible to formalise the constraints under which it *is* true?

This is a problem I've encountered when trying to generalize the property that each positive semidefinite matrix with trace 1 can be written as a convex combination of rank-one positive semidefinite terms $x x^*$ with trace 1. I was wondering how general it is and I did not manage to prove this.