Geometric boundary characterization by rank loci

Let $M$ be a matrix represented as a slack matrix $M=S_{P,Q}$ of nested polytopes $P\subseteq Q\subseteq\mathbb R^{r-1}$. A spectrahedral shadow $C$ of size $k$ between $P$ and $Q$ is a linear projection of a spectrahedron of size $k$. The rank-$s$ locus of $C$ consists of points whose lifts include a positive semidefinite matrix of rank $s$. Geometric boundary conjecture. The matrix $M$ lies on the boundary $\partial \mathcal M_{r,k}$ if and only if every spectrahedral shadow $C$ of size $k$ satisfying $P\subseteq C\subseteq Q$ contains $k+1$ vertices of $P$ at rank-one loci and touches $k+1$ facets of $Q$ at rank-$(k-1)$ loci. This is presented as the geometric version of the boundary characterization by rank-one factors, and the supplied text gives no resolution.