The non-$T_0$ geometric-realisation conjecture for finite spaces

Let $Y$ be a finite topological space, and let $|Y|_\omega$ denote its non-$T_0$ geometric realisation. Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the finite-space maps used above. The geometric-realisation conjecture. The canonical map

$$|Y|_\omega\longrightarrow Y$$

lies in

$$\left\{\mathsf{W}\to\mathsf{V},\,\mathsf{V}\to\{\bullet\}\right\}^{lr}.$$

This is proposed as an analogue of a theorem for perfectly normal spaces, after dropping perfect normality; the paper gives evidence but leaves the assertion conjectural.