Minimal fillings in general position are non-degenerate binary trees

A minimal filling of a finite metric space in general position is a tree whose internal vertices have degree $3$ and whose edges are non-degenerate.

General-position conjecture. Every minimal filling of a finite metric space in general position is a non-degenerate binary tree.

Minimal fillings of finite metric spaces can have degenerate edges or vertices of degree greater than $3$; the conjecture asserts that these phenomena disappear for a dense, generic class of metrics.