Cycle-network characterization conjecture for persistent path homology

Let a cycle network have $n$ nodes, with $n\in\mathbb{N}$. Let its 1-dimensional path persistence diagram (PPD) be the persistence diagram produced by 1-dimensional persistent path homology (PPH), and let 1-dimensional Dowker persistent homology denote the corresponding Dowker construction.

Cycle-network characterization conjecture. On cycle networks having any number of nodes, 1-dimensional PPH is isomorphic to 1-dimensional Dowker persistent homology. More specifically, the 1-dimensional PPD of a cycle network on $n$ nodes, $n\in\mathbb{N}$, contains exactly one off-diagonal point

$$\left(1,\left\lceil\frac{n}{2}\right\rceil\right).$$

This conjecture seeks a complete characterization of 1-dimensional PPH for cycle networks, paralleling the characterization of 1-dimensional Dowker persistent homology cited in the source. The supplied text reports experimental evidence but no proof or resolution.