Edge-count conjecture for minimal kinematics graphs

Let $[n]=\{1,2,\dots,n\}$, and call a graph on $[n]$ a minimal kinematics graph when its edge set is inclusion-maximal among the graphs realizing minimal kinematics. Minimal kinematics edge-count conjecture. Every minimal kinematics graph on $[n]$ has $3n-6$ edges. The conjecture is consistent with the authors' data and with the results cited as EPS; it extends the observed $2$-tree examples, but remains unproved in the supplied text.