Hyperdeterminantal boundary conjecture for real six-term decompositions of quartics

Let $f$ be a ternary quartic of real rank $6$ and signature $(3,3)$, $(4,2)$, or $(5,1)$. Let $\operatorname{VSP}(f)_\mathbb{R}$ be the real points of the variety of sums of powers, and let $\operatorname{SSP}(f)_\mathbb{R}$ be the semialgebraic subset parametrizing decompositions into six real points. Let $\overline{\operatorname{SSP}(f)_\mathbb{R}}$ denote its closure. Hyperdeterminantal boundary conjecture. The semialgebraic set $\overline{\operatorname{SSP}(f)_\mathbb{R}}$ is strictly contained in the variety $\operatorname{VSP}(f)_\mathbb{R}$. The conjecture concerns the existence of a non-real-decomposition boundary inside the real variety of sums of powers; the surrounding discussion identifies its Zariski closure with a hyperdeterminantal hypersurface, but does not establish that this boundary always exists. The source gives no resolution status.