  The Stanley-Stembridge conjecture associates a symmetric function to each
natural unit interval order $\mathcal P$. In this paper, we define relations
\`a la Knuth on the symmetric group for each $\mathcal P$ and conjecture that
the associated $\mathcal P$-Knuth equivalence classes are Schur-positive,
refining theorems of Gasharov, Brosnan-Chow, and Guay-Paquet. The resulting
equivalence graphs fit into the framework of D graphs studied by Assaf.
Furthermore, we conjecture that the Schur expansion is given by column-readings
of $\mathcal P$-tableaux that occur in the equivalence class. We prove these
conjectures for $\mathcal P$ avoiding two specific suborders by introducing
$\mathcal P$-analog of Robinson-Schensted insertion, giving an answer to a long
standing question of Chow.
