 Stationary-indexed club principle

For a stationary $S\subseteq {\omega }_1$, there are cofinal ladders $A_{\delta }\subset \delta$ of order type $\omega$, indexed by limit ordinals in $S$, such that every uncountable $X\subseteq {\omega }_1$ contains one of these ladders. This predicts a contained ladder, whereas stationary diamond principle predicts a whole initial segment.