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

For any $r$-uniform hypergraph $\mathcal{H}$ with $m$ ($\ge 2$) 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 $\mathrm{\mid }e\setminus e^{\mathrm{\prime }}\mathrm{\mid }$ among all pairs of distinct edges $e,e^{\mathrm{\prime }}$ in $\mathcal{H}$. We will show that if $r\ge 3$, $\rho (\mathcal{H})\ge 2$ and $m\ge \frac{\rho (\mathcal{H}{)}^3}{2}+1$, then $P_l(\mathcal{H},k)=P(\mathcal{H},k)$ holds for all integers $k\ge \frac{2.4(m-1)}{\rho (\mathcal{H})\log (m-1)}$.