The conjecture that non-SAGE certificates fail beyond the corollary's hypotheses

Let $\bm{A}$ be a matrix and let $obreak\mathcal{P}(\bm{A})$ be its associated polytope. Write $\operatorname{ext}\mathcal{P}(\bm{A})$ and $\operatorname{int}\mathcal{P}(\bm{A})$ for the sets of extreme points and interior points, respectively, and let $\mathsf{C_{SAGE}}(\bm{A})$ and $\mathsf{C_{NNS}}(\bm{A})$ denote the SAGE and nonnegativity cones. The matrix satisfies the hypothesis of the preceding corollary when $\mathcal{P}(\bm{A})$ is full dimensional with either at most one interior exponent, or $n+1$ extreme points and at most two interior exponents.

Non-SAGE separation conjecture. If every $\bm{a}_i$ lies in either $\operatorname{ext}\mathcal{P}(\bm{A})$ or $\operatorname{int}\mathcal{P}(\bm{A})$, but $\bm{A}$ does not satisfy the hypothesis of that corollary, then

$$\mathsf{C_{SAGE}}(\bm{A}) \neq \mathsf{C_{NNS}}(\bm{A}).$$

The conjecture proposes that, under the stated regularity condition, the equality of the SAGE and nonnegativity cones occurs only in the exceptional cases covered by the corollary. The supplied text does not state whether this conjecture has been resolved.