The chromatic equivalence class of K1,n,n+2

The chromatic equivalence class of a graph G is the set of graphs that have the same chromatic polynomial as G. We find the chromatic equivalence class of the complete tripartite graphs K1,n,n+2 for all n≥2. This partially answers a question raised in Chia and Ho (2009), which asks for the chromatic equivalence class of the graph K1,m,n where 2≤m≤n.