Kaad's pure-state separation conjecture for Hilbert $C^*$-modules

Let $\mathscr A$ be a $C^*$-algebra, let $\mathscr E$ be a Hilbert $C^*$-module over $\mathscr A$, and let $\mathscr L$ be a closed $\mathscr A$-submodule of $\mathscr E$. For a state $\omega$ on $\mathscr A$, write $\iota_\omega:\mathscr E\to\mathscr E_\omega$ for the canonical map to the Hilbert-space localization, and let $S(\mathscr A)$ denote the state space of $\mathscr A$. Kaad's pure-state separation conjecture. In this situation, there exist a pure state $\omega$ and an element $x_0\in\mathscr E$ such that $\iota_\omega(x_0)$ is not in the closure of $\iota_\omega(\mathscr L)$. In particular, there exists a pure state $\omega$ such that $\iota_\omega(\mathscr L)$ is not dense in $\mathscr E_\omega$, and hence

$$\iota_\omega(\mathscr L)^\perp\neq 0.$$

The conjecture concerns strengthening the corresponding separation theorem from states to pure states; the source context says that Kaad and Lesch provided a positive answer, so the conjecture is treated as solved.