Hamiltonian-reduction chart conjecture for framed torsion-free sheaves

Let $Z$ be the affine bundle constructed from the extension data, let $G$ be the acting group, and let

$$\mu=(\mu_a,\mu_b):Z\longrightarrow HH_0(\mathscr C)$$

be the resulting moment-map analogue. For $\zeta\in HH_0(\mathscr C)$, write $\mu^{-1}(\zeta)$ for its fiber, and let $\mathcal M(v;\mathcal F)$ denote the moduli stack of $\mathcal F$-framed objects on $\mathbb P_{[E/\Gamma](\Xi_\zeta)}$. Hamiltonian-reduction chart conjecture. The quotient stack

$$[\mu^{-1}(\zeta)/G]$$

is a chart of $\mathcal M(v;\mathcal F)$ containing all $\mathcal F$-framed torsion-free sheaves. For generic $\zeta$, the action of $G$ on $\mu^{-1}(\zeta)$ is free, and there is a $G$-equivariant line bundle on $\mu^{-1}(\zeta)$ whose GIT stability condition is equivalent to torsion-freeness. This would identify the framed moduli problem with a Hamiltonian/GIT quotient; the supplied text gives no proof or resolution.