Conjecture PC5 on hereditary maximum parsimony trees

Let $S:=f_1f_2\ldots f_n$ be a sequence of characters (an alignment) on a set $X$ of taxa, with $|X|=m$, and let $\mathcal{T}$ be a maximum parsimony tree for $S$. For a subset $Y\subseteq X$, write $S|_Y$ for the sequence restricted to the taxa in $Y$, and $\mathcal{T}|_Y$ for the tree restricted to $Y$. Conjecture PC5. For each $k=4,\ldots,m-1$, there exists a subset $Y$ of $X$ with $|Y|=k$ such that $\mathcal{T}|_Y$ is a maximum parsimony tree for $S|_Y$. This conjecture asks whether every maximum parsimony tree has most-parsimonious restrictions at every intermediate number of taxa; the cases of one, two, and three taxa are excluded because their unrooted trees are unique. The paper investigates this heredity property through examples of maximum parsimony trees and presents the conjecture as an open question.