Compactness characterization by nonsurjective locally contractive maps

Let $(X,\rho)$ be an infinite Polish ultrametric space. A map $\varphi:X\rightarrow X$ is locally contractive if, for every $x\in X$, there is a neighborhood on which distances from $x$ are strictly contracted by $\varphi$. Compactness characterization conjecture. The space $X$ is compact if and only if every locally contractive map $\varphi:X\rightarrow X$ is not surjective.

This conjecture seeks to characterize compact infinite Polish ultrametric spaces through the dynamical behavior of locally contractive self-maps. The preceding corollary establishes the nonsurjectivity conclusion for locally contractive maps on perfect compact ultrametric spaces, while the general equivalence stated here remains open.