This is a follow-up of an older question.

Let $H = (V, E)$ be a hypergraph such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardinal. We say that $c: V\to \kappa$ is a *weak coloring* if for all $e \in E$ the restriction $c|_e$ is not constant. We call $c$ a *strong coloring*, if for all $e \in E$ the restriction $c|_e$ is injective. We let $\chi_w(H)$ be the smallest cardinal such that there is a weak coloring to that cardinal, and we define $\chi_s(H)$ similarly for strong colorings.

Suppose that $V$ is infinite and that there is an integer $n\in\mathbb{N}$ such that $1<|e|\leq n$ for all $e\in E$. Is it possible that $\chi_w(H)$ is finite, but $\chi_s(H)$ is infinite?