In 2024, Hikita showed that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets expand with positive coefficients in the basis of elementary symmetric functions. This result resolved the long-standing Stanley–Stembridge conjecture. Finding a combinatorial interpretation of the $e$-coefficients remains a major open problem. In this talk I will define powerful and strong $P$-tableaux and conjecture that they give upper and lower bounds for the $e$-coefficients of chromatic symmetric functions. As evidence for these conjectures, we obtain combinatorial interpretations for various e-coefficients which live in between strong and powerful $P$-tableaux. Additionally, we show how Hikita’s theorem relates to strong $P$-tableaux and the Shareshian–Wachs inversion statistic.