The pure-state separation conjecture for Hilbert $C^*$-modules

Let $E$ be a Hilbert $C^*$-module over a $C^*$-algebra $\sA$, let $L\subset E$ be a closed convex subset, and let $x_0\in E\setminus L$. For each state $\go$ on $\sA$, let $E^\go$ be the localized Hilbert space and let $\iom:E\to E^\go$ be the canonical map. The pure-state separation conjecture. If $L$ is an $\sA$-submodule, then there exists a pure state $\go$ such that $\iom(x_0)$ is not in the closure of $\iom(L)$. In particular, there exists a pure state $\go$ such that

$$\iom(L)^\perp\ne\{0\}.$$

This is the pure-state strengthening of the separation theorem, which guarantees such a state without requiring it to be pure. The paper explicitly presents the pure-state assertion as a conjecture.