Equal-CSF unicyclic graphs with different degree sequences

Determine whether there exist two connected unicyclic graphs with the same chromatic symmetric function and different degree sequences when the common cycle length is greater than 3.

Background

The paper notes that equal chromatic symmetric functions already occur for pairs of non-isomorphic connected unicyclic graphs with a 3-cycle and an even number of vertices. The exhibited pairs nevertheless have the same degree sequence, number of non-trivial rooted trees, and number of internal edges. The authors explicitly leave unresolved whether equal-CSF pairs with different degree sequences occur for unicyclic graphs whose cycle length exceeds 3.