Ivanov–Tuzhilin half-perimeter formula for minimal fillings

Let ${\mathcal M}=(M,\rho)$ be a finite metric space. For a tree $G$ connecting $M$, let ${\mathcal O}(M,G)$ be the set of tours of $M$ with respect to $G$, and for each $\pi\in{\mathcal O}(M,G)$ define the half-perimeter by

$$p({\mathcal M},G,\pi)=\frac{1}{2}\sum_{x\in M}\rho\bigl(x,\pi(x)\bigr).$$

Let $\operatorname{mf}({\mathcal M})$ denote the weight of a minimal filling. Ivanov–Tuzhilin hypothesis. For an arbitrary metric space ${\mathcal M}=(M,\rho)$,

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

where the minimum is taken over all binary trees $G$ connecting $M$. The formula proposes a tour-based characterization of minimal filling weight; the source attributes it to A. Ivanov and A. Tuzhilin and gives no resolution.