A singleton-corona conjecture for asymptotic dimension structures

Let $X$ be a set equipped with a normal $T_1$ form $\omega$, and let $\partial(\omega)$ denote its corona and $LS(\omega)$ its associated large-scale structure. Assume

$$\dim(X\cup\partial(\omega))=0.$$

Singleton-corona conjecture. If the asymptotic dimension of $LS(\omega)$ does not equal $0$, then there is an unbounded subset $C$ of $X$ whose corona consists of exactly one point.

The preceding theorem proves the contrapositive under the additional hypothesis that $LS(\omega)$ is coarsely totally disconnected: under that hypothesis, dimension zero forces the asymptotic dimension of $LS(\omega)$ to equal zero. The conjectural assertion concerns the remaining case and is not resolved in the supplied text.