The reducibility conjecture for the real rank boundary of quaternary quartics

Let $S^4(V^*)$ be the space of quaternary quartic forms, let $\mathcal{R}_{4,4}$ denote the set of such forms of real rank at most $10$, and let $\partial_{alg}\mathcal{R}_{4,4}$ be its algebraic boundary. For a quartic $f$, let $\Omega(f)$ denote the corresponding variety of apolar decompositions, and let $C_{10,4}$ be the cone of quartics expressible as

$$\sum_{i=1}^{10}\lambda_i\ell_i^4,$$

with $\lambda_i\in\mathbb{R}_{\geq 0}$ and $\ell_i\in V^*$.

Reducibility conjecture. The real rank boundary $\partial_{alg}\mathcal{R}_{4,4}$ is reducible. The discriminant of $\Omega(f)$ is one of its components. Further, at least one more component comes from the components of the algebraic boundary of $C_{10,4}$.

The preceding discussion notes that quaternary quartics of signature $(9,1)$ with no real points on $\Omega(f)$ have rank at least $11$, and that an open subset of $Q_{4,4}$ consists of forms of real rank greater than $10$. The conjecture predicts that these phenomena yield distinct components of the algebraic real-rank boundary.