Compare list-color functions of uniform hypergraphs with their chromatic polynomials (II)

For any $r$-uniform hypergraph $\mathcal{H}$ with $m$ ($\geq 2$) edges, let $P(\mathcal{H},k)$ and $P_l(\mathcal{H},k)$ be the chromatic polynomial and the list-color function of $\mathcal{H}$ respectively, and let $\rho(\mathcal{H})$ denote the minimum value of $|e\setminus e'|$ among all pairs of distinct edges $e,e'$ in $\mathcal{H}$. We will show that if $r\ge3$, $\rho(\mathcal{H})\ge 2$ and $m\ge \frac{\rho(\mathcal{H})^3}2+1$, then $P_l(\mathcal{H},k)=P(\mathcal{H},k)$ holds for all integers $k\geq \frac{2.4(m-1)}{\rho(\mathcal{H})\log(m-1)}$.