The maximal-tour bound for minimal filling weight

Let ${\cal M}=(M,\rho)$ be a pseudo-metric space joined by a tree $G$. For a tour $\pi\in{\cal O}(G)$, let $p({\cal M},\pi)$ denote its corresponding perimeter, and let $\operatorname{mf}({\cal M})$ denote the weight of a minimal filling.

Maximal-tour conjecture. One has

$$\operatorname{mf}({\cal M})\le\max_{\pi\in{\cal O}(G)}p({\cal M},\pi).$$

This is presented as one of two conjectures implying a formula for the weight of a minimal filling of an arbitrary pseudo-metric space. The supplied text does not give a resolution.