Reducedness conjecture for the Poisson ideal defining the symmetric quotient Schubert variety

Let $K_0\backslash\mathcal{G}_0$ be the quotient under consideration, let $\lambda$ be a dominant coweight, let $\omega_i$ be the fundamental coweights for $1\leq i\leq n-1$, and set

$$r_i=\langle \omega_i,-w_0(\lambda)\rangle.$$

For each $i$, let $A_i^{(r)}$ and $B_i^{(r)}$ be the indicated functions on $K_0\backslash\mathcal{G}_0$, and let $\mathcal{S}_0^{\leq\lambda}$ denote the corresponding symmetric quotient Schubert variety. Reducedness conjecture. The ideal of $\mathcal{O}(K_0\backslash\mathcal{G}_0)$ which is Poisson generated by the functions $A_i^{(r)}$ for $1\leq i\leq n-1$ and $r>\langle\omega_i,-w_0\lambda\rangle$, together with $B_i^{(r_i+1)}$ whenever $r_i$ is even, is reduced, and hence is the ideal defining $\mathcal{S}_0^{\leq\lambda}$ inside $K_0\backslash\mathcal{G}_0$. The preceding discussion explains that taking the radical is essential in the existing statement, and conjectures that adding the indicated $B_i^{(r_i+1)}$ generators removes the resulting non-reducedness.