$e$-positivity of chromatic quasisymmetric functions of bounce number three via the inverse Kostka matrix

Let $G$ be the incomparability graph of a natural unit interval order $P$ on $[n]$ with the associated Dyck path $\mathbf{d}$. Let $X_G(\mathbf{x},q)$ be its chromatic quasisymmetric function. Shareshian and Wachs conjectured that $X_G(\mathbf{x},q)$ is $e$-positive. At $q=1$, this is the Stanley--S