The self-CSF conjecture for sufficiently large spiders

Let a spider be a tree with at most one vertex of degree greater than two, and let $X_T^T$ denote the self-chromatic symmetric function of a spider $T$. Spider self-CSF conjecture. The self-CSF distinguishes all spiders of sufficiently large size. The source presents this as a weaker version of the conjecture that the self-CSF distinguishes all trees; no resolution is given.