The twisted big J-function conjecture for GIT quotients

Let $E$ be a finite-dimensional ${\bf T}\times{\bf G}$-representation, let $\underline{E}=W\times_{\bf G}E$ be the induced ${\bf T}$-equivariant vector bundle on $[W/{\bf G}]$, and assume $R^1\pi_*f^*\underline{E}=0$ for every $\beta\in\mathrm{Eff}(W,{\bf G},\theta)$. Let $\tilde{\mathds{J}}^{\theta,\varepsilon,E}$ be the twisted quasimap $J$-function.

Twisted big $J$-function conjecture. The function $\tilde{\mathds{J}}^{\theta,\varepsilon,E}$ lies on the Lagrangian cone encoding the genus-zero, ${\bf T}$-equivariant, $\underline{E}|_{W/\!/{\bf G}}$-twisted Gromov–Witten theory of $W/\!/_{\theta}{\bf G}$ with Novikov ring $\Lambda_{\bf T}$.

This is a twisted analogue of the relationship between quasimap $J$-functions and Gromov–Witten theory. The supplied text gives no resolution status.