Invariant sums-of-squares induction conjecture

Let $V\subseteq\mathbb{R}^n$ be a real algebraic variety, let $G$ be a finite automorphism group of $\mathbb{R}[V]$, let $\mathcal{A}=\mathbb{R}[V]$, let $\mathcal{B}=\mathbb{R}[V]^G$ be the subalgebra of $G$-invariant elements, and let $p\colon\mathcal{A}\to\mathcal{B}$ be the canonical projection. Write $\sum\mathcal{A}^2$ for the cone of sums of squares in $\mathcal{A}$.

Invariant sums-of-squares induction conjecture. The induced quadratic module satisfies

$$\operatorname{ind}^{p}\bigl(\sum\mathcal{A}^2\cap\mathcal{B}\bigr)=\sum\mathcal{A}^2.$$

This asks whether sums of squares in the invariant subalgebra induce exactly the sums of squares in the full coordinate algebra. The statement is explicitly presented as open; the preceding proposition establishes an analogous result for positive polynomials, not for sums of squares.