of our conjecture is that higher decoupling relations of the form $j^{2r} = Q_{2r}(j^0,j^2,\dots,j^{n^2+3n-2})$ exist for all $r\geq\frac{1}{2} (n^2+3n)$. Hence $\cH(n)^{O(n)}$ has a minimal strong generating set $\{j^0, j^2, \dots, j^{n^2+3n-2}\}$, and in particular is a $\cW$-algebra of type $\cW(2,4,6,\dots, n^2+3n)$. By computer calculation, we prove our conjecture for $n=2$ and $n=3$, but we are unable to prove it in general.

By a fundamental result of Dong-Li-Mason [@DLM], $\cH(n)$ has a decomposition of the form $$\cH(n) \cong \bigoplus_{\nu\in H} L(\nu)\otimes M^{\nu},$$ where $H$ indexes the irreducible, finite-dimensional representations $L(\nu)$ of $O(n)$, and the $M^{\nu}$’s are inequivalent, irreducible, admissible $\cH(n)^{O(n)}$-modules. Since the Zhu algebra of $\cH(n)^{O(n)}$ is abelian, each $M^{\nu}$ above is a highest-weight module. For any reductive group $G\subset O(n)$, $\cH(n)^G$ is also a direct sum of irreducible, highest-weight $\cH(n)^{O(n)}$-modules. Using a classical theorem of Weyl, we show that there is a finite set of irreducible $\cH(n)^{O(n)}$-submodules of $\cH(n)^G$ whose direct sum contains an (infinite) strong generating for $\cH(n)^G$. Since $\cH(n)^{O(n)}$ is finitely generated, this shows that $\cH(n)^G$ is finitely generated as a vertex algebra. Finally, assuming our conjecture that $\cH(n)^{O(n)}$ is strongly finitely generated, we show that $\cH(n)^G$ is strongly finitely generated as well. Since our conjecture holds for 