Boundary characterization for positive matrices with size-2 psd factorizations

Let $\mathcal{M}_{3,2}^{p \times q}$ be the set of matrices admitting size-$2$ positive semidefinite factorizations. For a given size-$2$ psd factorization of $M$, the conditions b), c), and d) are respectively: $2$-infinitesimal rigidity, local rigidity, and global rigidity.

Size-2 boundary characterization conjecture. A positive matrix $M$ is on the topological boundary of $\mathcal{M}_{3,2}^{p \times q}$ if and only if any one of the conditions in the cited equivalence theorem is true.

The proposed characterization combines the rigidity equivalence for positive matrices with the preceding boundary results. Its proof is presented as a consequence that would follow from the converse boundary proposition; it is not established in the supplied text.