On a conjecture concerning the property of chromatic polynomials with negative variables

Let $G$ be a graph of order $n$ and $P(G,x)$ be the chromatic polynomial of $G$. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that $\frac{d^k}{d{x}^k}(\ln [(-1{)}^{n}P(G,x)])<0$ holds for all $k\ge 2$ and $x\in (-\mathrm{\infty },0)$. We prove this conjecture for all $k\ge 2$ and $x\le -6.66\mathrm{\Delta }k$, in which $\mathrm{\Delta }$ is the maximum degree of $G$.