SOS conjecture on linear ranks of polynomial mappings

Let $P(z)=(P^1(z),\ldots,P^q(z))$ be a polynomial mapping in $z=(z^1,\ldots,z^n)\in\mathbb C^n$, and let $A(z,\bar z)$ be a Hermitian polynomial satisfying

$$\lVert P(z)\rVert^2=A(z,\bar z)\lVert z\rVert^2.$$

If $r$ denotes the linear rank of $P(z)$, and $\kappa_0$ is the largest integer satisfying the defining inequality for $\kappa_0$, SOS Conjecture. Either

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

or there exists an integer $1\leq\kappa\leq\kappa_0<n$ such that

$$\sum_{i=0}^{\kappa-1}(n-i)=n\kappa-\frac{\kappa(\kappa-1)}{2}\leq r\leq \kappa n.$$

The conjecture concerns the possible gaps in linear ranks of polynomial sums-of-squares identities and is presented as the algebraic statement from which the HJY Gap Conjecture follows.