Cardinal-preserving extensions

A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that$\omega_2^L$is countable: {$X \in L \mid X \subseteq \omega_1^L$and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of$\omega_1^L$which in L are stationary. Is there a similar such result for subsets of$\ omega_2^L$? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are "solvable" in the above sense, as well as defining a notion of reduction between them.