Chromatic symmetric homology distinguishes the graph pairs in Figure 1

Let the pairs of graphs be those constructed in Figure 1 by adding horizontally symmetric planar graphs to the leftmost and rightmost vertices of the graph shown there. The resulting graphs have identical chromatic symmetric functions.

Chromatic symmetric homology distinction conjecture. Each such pair is distinguished by chromatic symmetric homology.

If true, this gives an infinite family of graph pairs that chromatic symmetric functions cannot distinguish but chromatic symmetric homology can. The claim is presented as conditional on the paper's earlier conjecture concerning the relevant graph automorphisms, so its resolution is not established here.