Cyclic interval conjecture for odd cycles

Let $m$ be odd and let $x\subseteq\mathbb{Z}_m$ with $x\notin\{\emptyset,\mathbb{Z}_m\}$. An arc $W$ is a cyclic interval in $\mathbb{Z}_m$. Write $C_m$ for the cycle graph on $m$ vertices, and let $k_{\min}(C_m)$ denote the minimum dimension of an isometric embedding of $C_m$ into an abelian group. The lemma asserts that there is an arc $W$ satisfying

$$|W\triangle x|<\min(|W|,m-|W|).$$

Cyclic interval conjecture. The displayed inequality holds for every odd $m$ and every nonempty proper subset $x\subseteq\mathbb{Z}_m$; consequently,

$$k_{\min}(C_m)=m-1$$

for every odd cycle $C_m$.

The inequality is the remaining case beyond the covering-arc regime and has been verified exhaustively for all odd $m\leq 17$. If true, it gives the naive-dimension lower bound for every odd cycle; the general case remains open.