Uniform-mass conjecture for cycles

Let $C_m$ be the cycle on $m$ edges, let $E(C_m)$ be its edge set, and let $\beta(C_m,1)$ be the optimization parameter defined in the paper. Uniform-mass conjecture for cycles. For all $m\geq 3$, $\beta(C_m,1)$ is achieved by the uniform distribution on $E(C_m)$. In particular,

$$\beta(C_m,1)=m^{-m}.$$

The paper explains that this is open in general; it considers the cases $m=5,6$ approachable, while its methods do not settle $m=7$.