The invariant partial ideal characterization conjecture for $C^*$-algebras

Let $\mathcal{A}$ be a $C^*$-algebra. A partial ideal assigns an ideal $\pi(V)$ to each commutative sub-$C^*$-algebra $V\subseteq\mathcal{A}$ compatibly with inclusions, and it arises from a total ideal $I\subseteq\mathcal{A}$ when $\pi(V)=I\cap V$. A partial ideal is invariant when, for every commutative sub-$C^*$-algebra $V\subseteq\mathcal{A}$ and every unitary $u\in\mathcal{A}$,

$$\operatorname{Ad}_u(\pi(V))=\pi(\operatorname{Ad}_u(V)).$$

Invariant partial ideal conjecture. A partial ideal of a $C^*$-algebra arises from a total ideal if and only if it is an invariant partial ideal. Consequently, the map $I\longmapsto\pi_I$ is a bijective correspondence between total ideals and invariant partial ideals.

This conjecture seeks to characterize exactly which partial ideals are induced by closed two-sided ideals, using invariance under inner automorphisms. The supplied text gives no resolution status.