Ebenfelt's SOS conjecture for ranks of prolonged sums of squares

Let $n \geq 2$, let $A(z,\bar{z})$ be a real polynomial, and let ${\rm{SOS}}_n$ denote the set of real polynomials expressible as sums of squared moduli of holomorphic polynomials. Write $R$ for the rank of $A(z,\bar{z})\|z\|^2$, and let $\kappa_0$ be the largest integer such that

$$\frac{\kappa_0(\kappa_0+1)}{2}<n.$$

SOS Conjecture. If $A(z,\bar{z})\|z\|^2\in {\rm{SOS}}_n$, then either

$$R\geq (\kappa_0+1)n-\frac{(\kappa_0+1)\kappa_0}{2}-1,$$

or there exists $\kappa\in\{0,1,2,\cdots,\kappa_0\}$ such that

$$\kappa n-\frac{\kappa(\kappa-1)}{2}\leq R\leq \kappa n.$$

The conjecture concerns the gaps in possible ranks under the positive-semidefiniteness condition. Huang's lemma proves that $R=0$ or $R\geq n$, so the claim holds for $n=2$; partial results are known for $n\geq 3$, including the left-hand bound when $A\in {\rm{SOS}}_n$.