General explanation and scope of MC-discriminant/PSD-boundary coincidence

Determine whether there is a simple general explanation for the coincidence between the matrix completion discriminant $D(G)$ and the algebraic boundary $\partial_a\mathscr C(G)$ of the positive semidefinite completion cone, and determine how far this equality extends beyond cycle-completable graphs.

Background

The paper establishes equality between the two varieties for cycle-completable graphs by independent formulas, but emphasizes that the equality is not immediate from their definitions. The matrix completion discriminant allows arbitrary complex completions and complex stresses, whereas the positive semidefinite algebraic boundary is derived from real positive semidefinite completions and stresses.

The example of $K_{3,3}$ shows that the equality fails in general, while the later example of $W_5$ shows that equality can nevertheless hold outside the cycle-completable class. These contrasting examples motivate the unresolved questions about the mechanism behind the coincidence and the maximal graph class for which it remains valid.