Minimal fillings possess exact tours

Let ${\cal M}=(M,\rho)$ be a finite pseudo-metric space and let ${\cal G}$ be a minimal filling of ${\cal M}$. An exact tour is a tour whose associated boundary paths realize the corresponding distances and whose total length equals the weight of ${\cal G}$.

Exact-tour conjecture. Every minimal filling possesses an exact tour.

Exact tours would provide a direct combinatorial certificate for the weight of a minimal filling and are used in the paper to relate minimal fillings to parametric fillings and tour lengths. The supplied passage gives an example of an unstable minimal parametric filling that nevertheless possesses an exact tour, but does not establish the conjecture in general.