Colorability implies bounded-degree disconnectedness of graph homomorphism spaces

Let $H$ be a graph that is $k$-colorable. For a finite graph $G$, let ${\rm Hom}(G,H)$ denote the space of graph homomorphisms from $G$ to $H$, with adjacency given by changing one vertex image at a time. Colorability-disconnectedness conjecture. There is some finite graph $G$ of maximum degree less than $k$ such that

$${\rm Hom}(G,H)\text{ is disconnected}.$$

This proposed analogue of the preceding long-range-action theorem would connect the chromatic number of $H$ with disconnection of homomorphism spaces on boards of bounded degree. The source gives no proof or status beyond presenting the assertion as something one should perhaps be able to prove.