Three-node equivalence conjecture for persistent path and Dowker homology

Let $(X,A_X)$ be a three-node dissimilarity network. Write 1-dimensional persistent path homology for the persistent homology produced by persistent path homology (PPH), and write 1-dimensional Dowker persistent homology for the corresponding Dowker construction.

Three-node equivalence conjecture. On three-node networks, 1-dimensional PPH is isomorphic to 1-dimensional Dowker persistent homology.

This conjecture proposes that the two persistence methods agree in dimension one on the smallest nontrivial asymmetric networks. It is based on experimental observations; the supplied text gives no resolution beyond the conjecture.