Pointed e-positivity conjecture for unit interval graphs

Let $m=m_1\cdots m_n$ satisfy $1\leq m_1\leq\cdots\leq m_n\leq n$ and $m_i\geq i$ for each $i$, and let $G(m)$ be the graph on $[n]$ with an edge $(i,j)$ exactly when $i<j\leq m_i$. Let $X_{G(m),1}$ be the pointed chromatic symmetric function with distinguished vertex $1$. A pointed symmetric function is pointed e-positive when it is positive in the pointed elementary-symmetric basis. Unit-interval pointed e-positivity conjecture. $X_{G(m),1}$ is pointed e-positive. This would provide a pointed analogue of the elementary-positivity phenomenon sought for incomparability graphs of $(\mathbf{3}+\mathbf{1})$-free posets; the supplied text does not state a proof or resolution.