Genus-zero quasimap S-operator wall-crossing conjecture

Let $W//{\bf G}$ be a GIT quotient, let $\Lambda$ be the Novikov ring, and let $\gamma\in H^*(W//{\bf G},\Lambda)$ be of the form $\gamma={\mathbb 1}+O(q)$. For $0+\leq\varepsilon_1<\varepsilon_2\leq\infty$, let $\tau^{\varepsilon_1,\varepsilon_2}_\gamma$ be the generalized string transformation defined by comparing the $z^{-1}$ terms of the corresponding $S$-operators. Genus-zero quasimap wall-crossing conjecture. For all such $\gamma$ and stability parameters,

$$S^{\varepsilon_1}_{\tau^{\varepsilon_1,\varepsilon_2}_\gamma({\bf t})}(z)(\gamma)=S^{\varepsilon_2}_{\bf t}(z)(\gamma).$$

This conjecture predicts that changing the quasimap stability condition is governed by a change of variables. Its special case with $\gamma={\mathbb 1}$ gives the mirror-map relation between quasimap and Gromov--Witten theories; it is proved in some cases later in the paper but is not established for all targets.