Maximum parsimony recovery conjecture for low-score binary 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 $k<\frac{n}{4}$. Maximum parsimony recovery conjecture. Then $T$ is the unique maximum parsimony tree for $A_k(T)$. This conjecture formalizes the expectation that maximum parsimony uniquely recovers the generating tree when the number of substitutions is sufficiently small; the paper studies this question locally in the NNI neighborhood and motivates the broader recovery claim.