Let $\Delta(\kappa, \mu)$ be the statement: "let $F$ be a family of cardinality $\kappa$ of sets of cardinality less than $\mu$. Then there is a family $G \subset F$ of cardinality $\kappa$ and a set $r$ such that $a \cap b=r$ for every $a,b \in G$".

We know that if $\kappa$ is a regular cardinal and $\lambda^{<\mu} < \kappa$, for every $\lambda < \kappa$ then $\Delta(\kappa, \mu)$ holds. My question is:

Does $\Delta(\kappa, \mu)$ for some regular uncountable $\kappa$ and some uncountable $\mu<\kappa$ imply some weak form of the GCH below $\kappa$?