  In 1993, Stanley and Stembridge conjectured that a chromatic symmetric
function of any $(3+1)$-free poset is $e$-positive. Guay-Paquet reduced the
conjecture to $(3+1)$- and $(2+2)$-free posets which are also called natural
unit interval orders. Shareshian and Wachs defined chromatic quasisymmetric
functions, generalizing chromatic symmetric functions, and conjectured that a
chromatic quasisymmetric function of any natural unit interval order is
$e$-positive and $e$-unimodal.
  For a given natural interval order, there is a corresponding partition
$\lambda$ and we denote the chromatic quasisymmetric function by $X_\lambda$.
The first author introduced local linear relations for chromatic quasisymmetric
functions. In this paper, we prove a powerful generalization of the
above-mentioned local linear relations, called a rectangular lemma, which also
generalizes the result of Huh,Nam and Yoo. Such a lemma can be applied to
describe explicit formulas for $e$-positivity of a chromatic symmetric function
$X_\lambda$ where $\lambda$ is contained in a rectangle. We also suggest some
conjectural formulas for $e$-positivity when $\lambda$ is not contained in a
rectangle by applying the rectangular lemma.
