Criterion of NP-hardness for properly posed problems

Let $({\cal P})$ be a properly posed problem, and let ${\cal S}^+_c$ denote the positive canonical dual feasible set; write $\operatorname{int}{\cal S}^+_c$ for its interior. Criterion of NP-hardness. The problem $({\cal P})$ is NP-hard if and only if

$$\operatorname{int}{\cal S}^+_c=\emptyset.$$

This conjecture connects the computational difficulty of the nonconvex problem with the geometry of its canonical dual feasible set. The supplied text attributes the proposal to Gao and collaborators, but gives no resolution, so its status remains open.