Suppose we are working in the constructible universe L (i.e. assume V = L)
Let $\kappa$ be a regular (infinite) cardinal, and for each $\alpha \in [\kappa, \kappa^+)$ let $\beta(\alpha)$ be the least such that in $L_{\beta(\alpha) + 1}$ there is a surjection $\kappa$ onto $\alpha$.
Let $S$ be a subset of $\kappa^+$
Let $D = \{ \alpha < \kappa^+ \mid S \cap \alpha \in L_{\beta(\alpha)} \wedge L_{\beta(\alpha)} \models S \cap \alpha$ is stationary $\}$
If $D$ is stationary, is $S$ necessarily stationary?
I think it is and you can do something with elementary chains and condensation to get an element of your club that is in S. Being stationary is downward absolute