Lorentzianity conjecture for graph-coloring generating polynomials

Let $G$ be a graph with vertex set $V$, and for each $v\in V$ let $S(v)$ be its list of available colors. Let $p_\varnothing$ be the polynomial recursively defined from the associated random-walk data, and let ${\cal C}_\varnothing$ denote the corresponding cone. The Lorentzianity conjecture. There is a constant $C>0$ such that, if

$$|S(v)|\geq \operatorname{deg}(v)+C$$

for every $v\in V$, then $p_\varnothing$ is ${\cal C}_\varnothing$-Lorentzian. This conjecture is presented as a consequence suggested by the paper's family of Lorentzian polynomials; the source gives no resolution status.