The planarity conjecture for non-rigid irreducible digital images

Let $X$ be a digital image, meaning a finite graph-like digital topological space, and let $C_n$ denote the cycle of $n$ points. Call $X$ non-rigid if it admits a continuous self-map homotopic to the identity that is not the identity, and irreducible if it has no proper image homotopy equivalent to it. Let the adjacency graph of $X$ be the graph whose vertices are the points of $X$ and whose edges join adjacent points.

Planarity conjecture. If $X$ is a non-rigid irreducible digital image that is not isomorphic to $C_n$, then the adjacency graph of $X$ is not planar.

The conjecture is motivated by the enumeration of small connected digital images: the known non-rigid irreducible examples other than cycles have nonplanar adjacency graphs. It proposes that planarity excludes all further non-rigid irreducible images, but no resolution is supplied here.