Global maximum parsimony recovery conjecture for low-score characters

Let $T$ be a binary phylogenetic $X$-tree with $|X|=n$. For a character alignment $A_k(T)$ consisting of the binary characters of parsimony score $k$ on $T$, let $k\leq\frac{n-3}{3}$. Global maximum parsimony recovery conjecture. Then $T$ is the unique maximum parsimony tree for $A_k(T)$. This is presented as slightly stronger than the preceding conjecture because it asserts unique recovery under a broader bound on $k$; the paper proves a corresponding result within the NNI neighborhood for a smaller range, while global recovery remains open beyond the cases noted in the discussion.