The 4-adjacency cycle conjecture for digital images

Let $X$ be a digital image in ${\mathbb Z}^2$ equipped with 4-adjacency, where two lattice points are adjacent when they differ by one in exactly one coordinate. Let $C_n$ denote the cycle of $n$ points. A digital image is non-rigid when it admits a nonidentity continuous self-map homotopic to the identity, and irreducible when it has no proper image homotopy equivalent to it.

4-adjacency cycle conjecture. The image $X$ is non-rigid and irreducible if and only if $X$ is isomorphic to $C_n$ for some $n>4$.

The claim isolates the apparent source of all non-rigid irreducible images in the 4-adjacency lattice: cycles. It is presented as plausible even independently of the planarity conjecture, and the supplied text gives no resolution.