The extremal rays of the PSD cone are the rank one matrices (also known as projections),so those of the form $x x^t.$ - see, for example, https://math.stackexchange.com/questions/678693/positive-semidefinite-cone-is-generated-by-all-rank-one-matrices

Your conjectured result follows immediately.

The extremal rays of the PSD cone are the rank one matrices (also known as projections),so those of the form $x x^t.$ - see, for example, https://math.stackexchange.com/questions/678693/positive-semidefinite-cone-is-generated-by-all-rank-one-matrices

The answer to your question follows immediately.