Equality of parsimony consistency threshold between rooted n-taxa and unrooted (n+1)-taxa cases

Establish whether the minimum internal branch length threshold x_min that guarantees statistical consistency of concatenated parsimony under the multispecies coalescent with an infinite-sites mutation model is identical for the rooted n-taxa case and the unrooted (n+1)-taxa case, for all integers n ≥ 5.

Background

The paper analyzes consistency and inconsistency of concatenated parsimony under the multispecies coalescent with an infinite-sites model. It derives a critical minimum internal branch length x_min ≈ 0.062205 (in coalescent units) that guarantees consistency for the rooted 5-taxa case when branch lengths meet or exceed this threshold.

In the unrooted 6-taxa case, the authors observe the same threshold x_min ≈ 0.062205 guarantees consistency. Based on this numerical agreement between rooted 5-taxa and unrooted 6-taxa, they conjecture a general pattern relating rooted n-taxa and unrooted (n+1)-taxa thresholds, motivating a broader theoretical investigation.

References

The numerical agreement of the x_min needed for consistency between the rooted 5-taxa and unrooted 6-taxa case is perhaps not terribly surprising: we conjecture that such a result holds true between the rooted $n$-taxa and unrooted $(n+1)$-taxa cases for all $n\geq 5$.

— Inconsistency of parsimony under the multispecies coalescent  (2407.02634 - Rickert et al., 2024) in Section 4.3 (The parsimony anomaly zone for the unrooted 6-taxa case)