Exactness conjecture for the image of the cohomological map

Let $T_2$ be the torus acting on $V_1$ with character $t_1$ and on $V^*$ with character $t_2$, where $t_1,t_2\in\mathfrak{t}_2^*$ are the projections onto the respective factors. Let $\overline{\xi}$ be the map whose image lies in the subspace of polynomials $P\in\operatorname{Sym}_{\mathrm{H}^*_{T_2}}(\mathfrak{h}^*)$ such that the restriction of $P$ to

$$t_1+\alpha=t_2-\beta=0\subseteq\mathfrak{h}_{\mathrm{H}^*_T}$$

vanishes for every pair of weights $\alpha,\beta$ of $V$ such that $\beta-\alpha$ is a weight of $\mathfrak{g}$. Exactness conjecture. The image of $\overline{\xi}$ is exactly this subspace of polynomials.