Conjecture on rank separations for polynomials in two variables

Let $p\in\mathbb{R}[x,y]$. The quantities $\operatorname{rank}_{\Lambda_1}(p)$, $\operatorname{sep\text{-}rank}_{\Lambda_1}(p)$, and $\operatorname{sos\text{-}rank}_{\Lambda_1}(p)$ denote, respectively, the rank, separable rank, and sum-of-squares rank of $p$, with separable rank taken with respect to the local sum-of-squares cones. The polynomial $p$ is understood to be separable in the first two cases and a sum of squares in the third.

Rank-separation conjecture. There exist no functions $f,g,h\colon\mathbb{N}\to\mathbb{N}$ such that, for every $p\in\mathbb{R}[x,y]$, independently of the degree of $p$,

$$\operatorname{sep\text{-}rank}_{\Lambda_1}(p)\leq f\bigl(\operatorname{rank}_{\Lambda_1}(p)\bigr),$$ $$\operatorname{sep\text{-}rank}_{\Lambda_1}(p)\leq g\bigl(\operatorname{sos\text{-}rank}_{\Lambda_1}(p)\bigr),$$

and

$$\operatorname{sos\text{-}rank}_{\Lambda_1}(p)\leq h\bigl(\operatorname{rank}_{\Lambda_1}(p)\bigr).$$

The conjecture asserts that rank measures for polynomials can exhibit unbounded separations even when the number of variables is fixed and the degree is unrestricted; the preceding constructions establish analogous separations for bounded-degree polynomials, but the two-variable, arbitrary-degree case remains posed as a conjecture.