If $H=(V,E)$ is a hypergraph and $\kappa \neq \emptyset$ is a cardinal, then a map $c:V\to\kappa$ is called a *coloring* if the restriction $c\restriction_e$ is non-constant for all $e\in E$ with $|e|\geq 2$. The smallest cardinal $\kappa$ for which a coloring $c:V\to \kappa$ exists, is said to be the *chromatic number* $\chi(H)$ of $H$.

We say $J\subseteq V$ is *independent* if $|J\cap e| \leq 1$ for all $e\in E$. We say that a collection ${\cal J}$ of independent sets is an *independent covering* of $H$ if $\bigcup {\cal J} = V$. The smallest cardinality of any independent covering of $H$ is said to be the *independent covering number* $j(H)$ of $H$.

It is not difficult to see that if $H$ is a graph, then $j(H) = \chi(H)$.

We say $H$ is $k$-regular for an integer $k\geq 2$ if $|e|=k$ for all $e\in E$.

**Question.** Given an integer $k\geq 3$ is there a positive integer $C_k$ such that
$$j(H) \leq C_k\cdot \chi(H)$$ for all finite $k$-regular hypergraphs $H$?