Open quantum Kirwan map conjecture for disk vortices

Let $(X,\omega,\mu)$ be a Hamiltonian $G$-manifold, let $\bar{X}=\mu^{-1}(0)/G$, and let $L\subset\mu^{-1}(0)$ be the $G$-invariant lift of a Lagrangian $\bar{L}$. Let $\mathcal{QF}(L)$ be the quasimap $A_\infty$ algebra and $\mathcal{Fuk}(\bar{L})$ the Fukaya algebra. Let $q\kappa_0^X:\Lambda\to QH(\bar{X})$ be the zeroth component of the closed quantum Kirwan map $q\kappa^X:QH^G(X)\to QH(\bar{X})$, and set

$$a_X=q\kappa_0^X(1)\in QH(\bar{X}).$$

Open quantum Kirwan map conjecture. Counting affine vortices over the upper half-plane $\mathbb H$ defines an $A_\infty$ morphism

$$q\kappa^{X,L}:\mathcal{QF}(L)\to\mathcal{Fuk}(\bar{L};a_X),$$

where $\mathcal{Fuk}(\bar{L};a_X)$ is the bulk deformation of $\mathcal{Fuk}(\bar{L})$ by $a_X$. The morphisms obtained from different choices of data are $A_\infty$-homotopic, and the gauged and bulk-deformed correlation functions satisfy

$$\bar\tau^{X,L}_{\underline{k}}\bigl(a_X;q\kappa^{X,L}(b_1),\ldots,q\kappa^{X,L}(b_{\underline{k}})\bigr)=\lim_{\lambda\to+\infty}\tau^\lambda_{\underline{k}}(b_1,\ldots,b_{\underline{k}}),$$

possibly up to $A_\infty$ homotopy.

This is the open analogue of the quantum Kirwan map: it is intended to relate disk-vortex counts to the Fukaya-theoretic correlation function of the reduced Lagrangian in the adiabatic limit. The source presents this as a conjectural construction and does not state a resolution.