Draisma's topological noetherianity conjecture for the affine infinite Grassmannian

Let $k$ be a field. Define the affine infinite Grassmannian $\widetilde{\operatorname{Gr}}(\infty/2,V^*_\infty):=\varprojlim_{(n,p)}\widetilde{\operatorname{Gr}}(p,V^*_{n,p})$, and let $S_\infty$ be the infinite symmetric group acting on it by permutation matrices. Let $\operatorname{N}(\infty)$ be the direct-limit subgroup formed by the normalizers of the standard maximal tori.

Draisma's conjecture. The affine infinite Grassmannian $\widetilde{\operatorname{Gr}}(\infty/2,V^*_\infty)$ is topologically $S_\infty$-noetherian: every descending chain of $S_\infty$-stable Zariski-closed subsets stabilizes. A weaker form asserts topological $\operatorname{N}(\infty)$-noetherianity, while a stronger form asserts that every ascending chain of $S_\infty$-stable ideals in its affine coordinate ring stabilizes.

The paper disproves the stronger form by exhibiting an ascending chain of $S_\infty$-stable ideals that does not stabilize. Thus the topological conjecture and its weaker $\operatorname{N}(\infty)$-form remain unresolved in the supplied text.