Conjecture on structural data determined by the CSF

Prove that if two connected unicyclic graphs, each either having a 3-cycle and an odd number of vertices or having a cycle of length at least four, possess identical chromatic symmetric functions, then they have the same degree sequence, the same number of non-trivial rooted trees, and the same number of internal edges.

Background

The paper reports that connected unicyclic graphs with a 3-cycle and an even number of vertices can occur in non-isomorphic pairs with identical chromatic symmetric functions while having different structural data. In contrast, the authors observe computationally that the proposed rigidity statement holds for the specified classes through 16 vertices.

The conjecture formalizes the observed restriction for 3-cycles of odd order and for cycles of length at least four. It asks for a proof that equality of the chromatic symmetric function determines the degree sequence, the number of non-trivial rooted trees, and the number of internal edges in those cases.