The nontrivial-invariant analogue of Valiant's conjecture

Let $\mathbf{k}=\mathbb{R}$ or $\mathbb{C}$, and let $K_0=K_0(\operatorname{sa}_\mathbb{R})$ when $\mathbf{k}=\mathbb{R}$ and $K_0=K_0(\operatorname{Var}_\mathbb{C})$ when $\mathbf{k}=\mathbb{C}$. Let $\mathrm{inv}:K_0\rightarrow\mathbf{A}$ be a non-trivial additive invariant, where $\mathbf{A}$ is a polynomially bounded graded $\mathbf{k}$-algebra. Analog of Valiant's conjecture.

$$\mathbf{VP}_{\mathbf{k}}^{\mathbf{A}}\neq\mathbf{VNP}_{\mathbf{k}}^{\mathrm{inv}}.$$

This is the constructible-function analogue of Valiant's $\mathbf{VP}$ versus $\mathbf{VNP}$ conjecture; the source presents it as an open separation question for every non-trivial additive invariant.