Completeness conjecture for physically relevant SO(3)-invariant Jordan multialgebras

Let $\mathcal T$ be a real finite-dimensional representation of $SO(3)$ of the form

$$\mathcal T=W_{0}\otimes\mathbb R^{n_{0}}\oplus W_{1}\otimes\mathbb R^{n_{1}}\oplus W_{2}\otimes\mathbb R^{n_{2}},\qquad n_{0},n_{1},n_{2}\geq 0,$$

where $W_{0}$, $W_{1}$, and $W_{2}$ are respectively the weight $0$, natural, and symmetric trace-free representations. Let $\mathcal A\subset\operatorname{Sym}(\mathcal T)$ be isomorphic to $W_{2}\oplus W_{4}$ as an $SO(3)$-module. Completeness conjecture. Any rotationally invariant $\mathcal A$-multialgebra is complete. This concerns the completeness of the Jordan multialgebras arising in physically relevant coupled field problems; the paper notes supporting results for $SO(3)$-invariant examples, while the general claim remains open.