NNI-sequence conjecture for tree topologies along tropical line segments

Let $T_1$ and $T_2$ be equidistant rooted phylogenetic trees with $n$ leaves in $X$, and let $\Gamma_{T_1,T_2}$ be the tropical line segment between them. A nearest neighbor interchange (NNI) move is the local tree-topology transformation used to relate the topologies of $T_1$ and $T_2$. NNI-sequence conjecture. Given the tree topologies of $T_1$ and $T_2$, the tree topology changes according to a sequence of NNI moves from $T_1$ to $T_2$ along $\Gamma_{T_1,T_2}$. This conjecture seeks to describe the combinatorics of tree topologies along tropical line segments between equidistant trees; the paper presents it as an open problem for general $n$.