The Stanley–Stembridge conjecture for chromatic symmetric functions

Let $P$ be a finite partially ordered set, and let $G$ be its incomparability graph. For a proper coloring of $G$, the associated chromatic symmetric function is $X_G$. A symmetric function is $e$-positive if its expansion in the basis of elementary symmetric functions has nonnegative coefficients.

Stanley–Stembridge conjecture. If $P$ is a ${\bf 3}+{\bf 1}$-avoiding partial order, then $X_G$ is $e$-positive.

The conjecture was reduced to natural unit interval orders by Guay-Paquet and was proved by Brosnan and Chow, independently of Guay-Paquet. The paper studies and reproves the special case for unit interval orders additionally avoiding ${\bf 2}+{\bf 1}+{\bf 1}$, originally due to Gebhard and Sagan.