Stability 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. A polynomial is stable if it has no roots in the product of open upper half-planes. Stability conjecture. The resulting polynomial is stable. The source describes this as a conjecture based on computational experiments and notes that stability would strengthen the Lorentzianity conjecture; it remains open.