Cubic generation conjecture for path phylogenetic ideals

Let $T$ be a path of length $n$, and let $I_T$ be its toric ideal of phylogenetic invariants. The ideal $I_T$ should be generated in degree $3$, with $2n-4$ generators of degree $3$ necessary. The conjecture is based on the tabulated Markov-basis data for paths; no proof or resolution is given in the source.