Homotopy triviality of digital images without simple loops of length at least five

Let $X$ be a finite digital image, viewed as a graph, and let a simple $m$-loop mean a simple loop in $X$ with $m$ vertices. Homotopy triviality conjecture. If $X$ has no simple $m$-loop for any $m\ge 5$, then $X$ is homotopy equivalent to a point. This strengthens the preceding theorem for images with no simple $m$-loops for $m\ge 4$; the authors state that the claim seems true but that they have been unable to prove it.