Suppose $\kappa$ is measurable with a witnessing normal ultrafilter $U$ and $P$ is a ccc forcing. Let $\langle p_i: i < \kappa \rangle$ be a sequence of conditions in $P$ such that for every $X \in U$, $\{p_i: i \in X\}$ is predense in $P$. Must there exist an infinite $X \subseteq \kappa$ such that for every infinite $Y \subseteq X$, $\{p_i: i \in Y \}$ is predense in $P$?

$D$ is predense in $P$ means: Every condition in $P$ is compatible with some condition in $D$.