The polynomial semiring sum-product partition conjecture

Let $\mathbb{N}[x]$ be the polynomial semiring with coefficients in $\mathbb{N}$, and let $\mathbb{N}[x]=C_0\cup C_1$ be a bipartition. The polynomial semiring conjecture. There exist $i\in\{0,1\}$ and $f,g\in\mathbb{N}[x]$ such that

$$\{f,g,f+g,fg\}\subseteq C_i.$$

This asks for a monochromatic additive-multiplicative configuration in the polynomial semiring, where the paper identifies $\mathbb{N}[x]$ as an example not covered by its preceding subsemiring method; the source gives no resolution.