The invariant partial ideal conjecture for C*-algebras

Let $\mathcal{A}$ be a $C^*$-algebra. A partial ideal assigns an ideal $\pi(V)$ to each commutative $C^*$-subalgebra $V\subset\mathcal{A}$, compatibly with inclusions. It is invariant when, for every unitary $u\in\mathcal{A}$ and every such $V$,

$$u\pi(V)u^*=\pi(uVu^*).$$

The invariant partial ideal conjecture. A partial ideal of $\mathcal{A}$ arises from a total ideal if and only if it is an invariant partial ideal. Consequently, the map $I\mapsto\pi_I$ is a bijective correspondence between total ideals and invariant partial ideals. This conjecture characterizes which contextwise ideal data come from genuine closed, two-sided ideals; the supplied text gives no resolution, while related structural results are established for von Neumann algebras.