Degree-sequence and structural invariants conjecture for chromatic symmetric functions of unicyclic graphs

Let $G_1$ and $G_2$ be connected unicyclic graphs, where either each has a 3-cycle and an odd number of vertices, or each has a $c$-cycle for some $c\geq4$. Let $\mathbf{X}_G$ denote the chromatic symmetric function of a graph $G$. Unicyclic graph invariants conjecture. If

$$\mathbf{X}_{G_1}=\mathbf{X}_{G_2},$$

then $G_1$ and $G_2$ have the same degree sequence, the same number of non-trivial rooted trees, and the same number of internal edges. The conjecture concerns which structural features of connected unicyclic graphs are determined by their chromatic symmetric functions; it had been verified for graphs with $n\leq16$ vertices in the source, while the contrasting 3-cycle examples with an even number of vertices show why the parity restriction is necessary.