The question is concerned with symmetric matrices $\mathbb{S}\_n$ as a real vector space. Let $X$ be a positive semidefinite symmetric matrix, and let $P : \mathbb{S}\_n \to \mathcal{V}$ be a projection onto some subspace $\mathcal{V} \subset \mathbb{S}\_n$. Is it always the case that $\mathrm{trace}(P(X)) \leq \mathrm{trace}(X)$?

I can see this being true when $\mathcal{V}$ has an orthonormal basis $\{A\_i\}$ consisting of matrices which are all either trace $0$ or positive and trace $\leq 1$. But in general I don't know anything about what kind of basis $\mathcal{V}$ would admit. Is it possible to come up with a counterexample?