Triangle inequality for PLR on binary gene trees

Establish whether the Path-Label Reconciliation (PLR) dissimilarity satisfies the triangle inequality when restricted to binary reconciled gene trees, thereby determining whether PLR is a metric on that subclass.

Background

The paper proves that PLR is a semi-metric under least duplication-resolved equivalence and shows that PLR violates the triangle inequality for non-binary gene trees. Empirically, the authors did not find counterexamples among binary trees, leaving open whether PLR might satisfy the triangle inequality in that restricted setting.

Resolving this would clarify the theoretical status of PLR for common phylogenetic scenarios where gene trees are binary, potentially elevating PLR from a semi-metric to a metric in that domain.