The proper-map characterization by the simplest finite counterexample

Let $\{o\}\to\{o\to c\}$ be the indicated map of finite topological spaces, and let the subscript $<5$ denote the subclass consisting of maps between spaces with fewer than $5$ points. The simplest-counterexample conjecture.

$$\left(\left\{\{o\}\to\{o\to c\}\right\}^{r}_{<5}\right)^{lr}$$

is the class of proper maps. The preceding calculation identifies the finite-space subclass with proper maps on spaces of size less than $5$, while the conjectured global characterization remains open.