The ideal conjecture for affine-Grassmannian slices at the base point

Let $\Gr^{\overline{\lambda}}_0$ be the slice at $\mu=0$ inside $\Gr_0=G[[t^{-1}]]_1$, and let $J^\lambda_0$ be the Poisson ideal of $\mathcal O(\Gr_0)$ generated by the coefficients $\Delta_{\omega_i,\omega_i}^{(s)}$ for $s>\langle\lambda,\omega_{i^*}\rangle$ and all $i\in I$. The ideal conjecture. The ideal of $\Gr^{\overline{\lambda}}_0$ in $\mathcal O(\Gr_0)$ is $J^\lambda_0$. This gives an explicit Poisson-algebraic description of the slice and is part of the paper's conjectural presentation of its defining ideal; no proof or disproof is supplied here.