The tour formula for minimal filling weight

For a pseudo-metric space ${\cal M}=(M,\rho)$, let $\operatorname{mf}({\cal M})$ be the weight of a minimal filling. For a tree $G$ with boundary $M$, let ${\cal O}(G)$ be its tours and let $p({\cal M},\pi)$ be the perimeter associated with a tour $\pi$.

Minimal-filling tour formula. For an arbitrary pseudo-metric space ${\cal M}=(M,\rho)$,

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

where the minimum may be taken over all trees with boundary $M$, or equivalently over all binary trees with boundary $M$.

The paper says this formula follows from the two preceding conjectures, so it is a conjectural consequence rather than an independently established theorem in the supplied passage. It would give a formula for calculating the weight of a minimal filling for every pseudo-metric space.