Dong–Ge–Gong–Ning–Ouyang–Tay higher-derivative conjecture for chromatic polynomials

Let $G$ be a graph of order $n$, and let $P(G,x)$ denote its chromatic polynomial. For $x<0$, the quantity $(-1)^nP(G,x)$ is positive, so its logarithm is defined. Dong–Ge–Gong–Ning–Ouyang–Tay's higher-derivative conjecture. The inequality

$$\frac{d^k}{dx^k}\left(\ln[(-1)^nP(G,x)]\right)<0$$

holds for all $k\geq 2$ and $x\in(-\infty,0)$. This extends the known negativity of the first derivative, $P'(G,x)/P(G,x)<0$, to all higher derivatives and concerns the sign pattern of logarithmic derivatives of chromatic polynomials on the negative real axis; its resolution is not stated here.