  Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform
hypergraph with $m$ edges. By refining the broken cycle theorem for
hypergraphs, we show that if $k>\frac{m-1}{\ln(1+\sqrt{2})}\approx 1.135 (m-1)$
then the $k$-list assignment of $G$ admitting the fewest colorings is the
constant list assignment. This extends the previous results of Donner,
Thomassen and the current authors for graphs.
