Cox and Martin's weighted optimization conjecture for even-length paths

Let $U$ be a finite set, and let $\mu$ be a probability measure on the edges of the complete graph on $U$, so that $\mu(uv)\ge 0$ and

$$\sum_{\{u,v\}\in\binom{U}{2}}\mu(uv)=1.$$

For $u\in U$, define the weighted degree by

$$\mu(u)=\sum_{v\in U\setminus\{u\}}\mu(uv).$$

For every integer $\ell\ge 2$, set

$$\rho(\mu;\ell)=\sum_{\substack{(u_1,\ldots,u_\ell)\in U^\ell\\ u_1,\ldots,u_\ell\ \operatorname{pairwise\ distinct}}}\mu(u_1)\left(\prod_{i=1}^{\ell-1}\mu(u_i u_{i+1})\right)\mu(u_\ell),$$

and define

$$\rho(\ell)=\sup_{U,\mu}\rho(\mu;\ell),$$

where the supremum is over all finite sets $U$ and all such probability measures. Cox and Martin's conjecture. For every integer $\ell\ge 2$,

$$\rho(\ell)=\frac{8}{\ell^\ell}.$$

This weighted optimization parameter controls the leading term in the upper bound for the number of even-length paths in planar graphs. The source attributes the conjecture to Cox and Martin and does not provide evidence of its resolution, so its general status remains open.