Face-relative-interior conjecture for stubborn forms

Let $X\subset \mathbb{P}^n$ be a totally real variety, let $P_X$ denote its cone of nonnegative forms, and let $\mathcal{F}$ be a face of $P_X$. Face-relative-interior conjecture. Either every point in the relative interior of $\mathcal{F}$ is stubborn, or no point in the relative interior of $\mathcal{F}$ is stubborn. For smooth curves, this follows from the characterization in terms of real zeroes, but the assertion is open for general totally real varieties.