Let $H=(V,E)$ be a hypergraph. If $\kappa>0$ is a cardinal, we say the hypergraph $H$ is $\kappa$-chromatic if there is a function $c:V\to\kappa$ such that for all $e\in E$ the restriction $c|_e$ is not constant (that is, the vertices of every edge are colored with at least $2$ colors).
We say that $H$ is a clique hypergraph if for all $e,f \in E$ we have $e\cap f \neq \emptyset$ and we call $H$ $\kappa$-regular for some cardinal $\kappa$, if $|e| = \kappa$ for all $e\in E$.
Question. If $\kappa \geq \aleph_0$ and $H=(V,E)$ is a $\kappa$-regular clique hypergraph that is $3$-chromatic, but not $2$-chromatic, are there $e\neq f\in E$ such that $|e\cap f| = \kappa$?