The topological extension conjecture for total ideals of $C^*$-algebras

Let $G(\mathcal{A})$ denote the geometric object associated to a $C^*$-algebra $\mathcal{A}$ by the extension procedure described in the source. Let $\mathscr{T}:\operatorname{KHaus}\longrightarrow\operatorname{CMSLat}$ assign to each compact Hausdorff space its complete lattice of closed sets, and let $\widetilde{\mathscr{T}}$ be its extension to $C^*$-algebras. Let $\mathcal{I}$ denote the functor assigning the relevant lattice of ideals.

Topological extension conjecture. The functors $\widetilde{\mathscr{T}}$ and $\mathcal{I}$ are naturally isomorphic.

This formalizes the conjecture that extending the notion of an open or closed subset from spaces to noncommutative geometric objects yields the notion of a closed, two-sided, or total, ideal of a $C^*$-algebra. The supplied text gives no resolution status.