Quantum Kirwan map conjecture in the semipositive case

Let $(M,\omega,\mu)$ satisfy (H), be convex at infinity, and be semipositive. Let $\varphi$ be a map

$$\varphi:H_2^G(M,\mathbb{Z})\to\operatorname{Hom}_{\mathbb{Q}}\big(H_G^*(M),H^*({\overline{M}})\big)$$

satisfying the equivariant Novikov condition, and let $\varphi_*$ be the induced map

$$\varphi_*:{\operatorname{QH}}^*_G(M,\omega)\to{\operatorname{QH}}^*({\overline{M}},{\overline{\omega}}).$$

Here $\kappa$ denotes the classical Kirwan map. Quantum Kirwan map conjecture. There exists such a map $\varphi$ for which $\varphi_*$ is a surjective ring homomorphism and

$$\varphi(0)=\kappa,$$

with

$$\big\langle[\omega-\mu],B\big\rangle\leq 0,\ B\neq 0\quad\Longrightarrow\quad\varphi(B)=0.$$

If proven, this would give a recursion formula for the quantum cohomology of the symplectic quotient in terms of equivariant quantum cohomology and $\varphi$.