The removable-edge conjecture for surjective polymorphisms of reflexive cycles

Let $n\geq 4$, let $\mathbb{G}$ be an $n$-cycle, and let $f$ be a surjective homomorphism from a product of subpaths of $\mathbb{G}$ of length $n-2$ to $\mathbb{G}$. An edge $e$ of $\mathbb{G}$ is removable for $f$ if

$$f(e')\neq e$$

for every edge $e'$ of the product. Removable-edge conjecture. There exists a removable edge of $\mathbb{G}$ for $f$; equivalently, there exists an edge of $\mathbb{G}$ that can be removed while $f$ remains a homomorphism. This conjecture concerns the structure of surjective polymorphisms of reflexive cycles and the possibility of reducing the target cycle without destroying the homomorphism property. The supplied text gives no evidence of resolution.