  In 2004 Pachter and Speyer introduced the higher dissimilarity maps for
phylogenetic trees and asked two important questions about their relation to
the tropical Grassmannian. Multiple authors, using independent methods,
answered affirmatively the first of these questions, showing that dissimilarity
vectors lie on the tropical Grassmannian, but the second question, whether the
set of dissimilarity vectors forms a tropical subvariety, remained opened. We
resolve this question by showing that the tropical balancing condition fails.
However, by replacing the definition of the dissimilarity map with a weighted
variant, we show that weighted dissimilarity vectors form a tropical subvariety
of the tropical Grassmannian in exactly the way that Pachter--Speyer
envisioned. Moreover, we provide a geometric interpretation in terms of
configurations of points on rational normal curves and construct a finite
tropical basis that yields an explicit characterization of weighted
dissimilarity vectors.
