Anisotropic extension of the Walpole–Kelvin equivalence criterion

Let $G$ be a subgroup of $\mathrm{SO}(3)$, let $\mathbb{T}$ be a tensor space with its $G$-representation, and let $\mathrm{C}\in\left[\mathbb{T}\otimes^S\mathbb{T}\right]^G$. Write $\sharp^G(\mathbb{T})$ for the number of $G$-irreducible components in the decomposition of $\mathbb{T}$, and regard $\left[\mathbb{T}\otimes^S\mathbb{T}\right]^G$ as the space of $G$-invariant symmetric endomorphisms of $\mathbb{T}$. Anisotropic Walpole–Kelvin equivalence conjecture. The Walpole and Kelvin representations of $\mathrm{C}$ are equivalent if and only if

$$\dim\left(\left[\mathbb{T}\otimes^S\mathbb{T}\right]^G\right)=\sharp^G(\mathbb{T}).$$

This conjectures that the criterion proved for isotropic tensors extends to anisotropic situations governed by a subgroup $G$ of $\mathrm{SO}(3)$. The paper does not investigate this extension, so its validity remains open.