Lorentzianity conjecture for chromatic symmetric functions of Dyck paths

Let $d$ be a Dyck path, let $G(d)$ be its indifference graph, and let $X_{G(d)}$ be its chromatic symmetric function. For any finite number of variables, restrict $X_{G(d)}$ to those variables. Lorentzianity conjecture. The resulting polynomial is Lorentzian. The conjecture was verified in the source for abelian Dyck paths and computationally for all Dyck paths of length $n\leq 7$ with at most $8$ variables; it remains open in general.