Flatness conjecture for fixed-point algebras of general quiver Coulomb branches

Let $I=(I_0,I_1)$ be a finite quiver with dimension vector ${\bf v}$ and framing ${\bf w}$, let $A$ be the natural torus acting on $\mathfrak{M}({\bf v},{\bf w})$, let $G_{\bf v}$ be the gauge group, and let $\mathcal{R}_{{\bf v},{\bf w}}$ be the corresponding variety of triples. Set $\mathbb{T}=(\mathbb{C}^\times)^{|I_0|}$. Flatness conjecture. The algebra

$$\mathbb{C}\left[\operatorname{Spec}\left(H_*^{A\times (G_{\bf v})_{\mathcal{O}}}(\mathcal{R}_{{\bf v},{\bf w}})\right)^{\mathbb{T}}\right]$$

is flat over $\mathbb{C}[\mathfrak{a}]$. This conjecture is proposed as a possible generalization of the paper's approach to other quivers; the supplied text gives no proof or resolution.