Uniform-mass conjecture for the path optimization parameter

For an integer $m\geq 2$, let $C_m$ be the cycle on $m$ edges, let $E(C_m)$ be its edge set, and let $\rho(m)$ be the path optimization parameter defined in the paper. Uniform-mass conjecture for $\rho(m)$. For all $m\geq 2$, $\rho(m)$ is achieved by the uniform distribution on $E(C_m)$. In particular,

$$\rho(m)=8\cdot m^{-m}.$$

This would imply the stated asymptotic formula for copies of $P_{2m+1}$ in planar graphs, with an $O(n^{m+4/5})$ error term. The paper proves only the cases $m=2$ and $m=3$, and notes that its methods are insufficient for the conjectured $O(n^m)$ error term.