Spectrahedron characterization of psd rank for normalized matrices

Let $M\in\mathbb R^{p\times q}_{\geq 0}$ have rank $k+1$ and satisfy

$$M\mathbf 1=\mathbf 1.$$

Let $P$ and $Q$ be the two polytopes corresponding to $M$ in the nested-polytope representation. Spectrahedron nesting conjecture. The matrix $M$ has psd rank at most $k$ if and only if a spectrahedron of size $k$ can be nested between $P$ and $Q$. The preceding lemma proves this equivalence in a restricted matrix-size setting; the conjecture asserts that it holds for matrices of any size.