Localization of symplectic quotient invariants to subtori of fixed dimension

Let $T$ be a torus and let $\leq i\leq \dim T$. For each subtorus $\tau\subset T$, let $\mathcal{A}_\tau$ denote the corresponding summand of the module $\mathcal{A}$, and set

$$\mathcal{A}_i=\bigoplus_{\dim\tau=i}\mathcal{A}_\tau.$$

Let $\mathcal{R}$ be the submodule of relations in $\mathcal{A}$. Localization conjecture. For any $0\leq i\leq \dim T$, the quotient $\mathcal{A}/\mathcal{R}$ is generated by the image of $\mathcal{A}_i$, equivalently by $\mathcal{A}_i/(\mathcal{R}\cap\mathcal{A}_i)$.

This conjecture strengthens the preceding proposition, which asserts generation using the summand associated with the full torus $T$. It asks whether the same generation property holds for every fixed subtorus dimension.