e-positivity conjecture for a three-parameter natural unit interval order

Let $P(\mathbf{m})$ be the natural unit interval order associated with a sequence $\mathbf{m}=(2,m_2,m_3,n,\ldots,n)$, and let $G$ be its incomparability graph. Let $\widetilde{X}_G(t)$ be the chromatic quasisymmetric function of $G$, and let $X_G=\widetilde{X}_G(1)$ be its chromatic symmetric function. Three-parameter e-positivity conjecture. The function $\widetilde{X}_G(t)$, and consequently $X_G$, is $e$-positive.

The claim concerns a specific family of natural unit interval incomparability graphs and is presented as a conjecture in the source. Its resolution status is not specified there.