Mixed Levels of Indestructibility

Starting from a supercompact cardinal $\kappa $, we force and construct a model in which $\kappa $ is both the least strongly compact and least supercompact cardinal and $\kappa $ exhibits mixed levels of indestructibility. Specifically, $\kappa $'s strong compactness, but not its supercompactness, is indestructible under any $\kappa $-directed closed forcing which also adds a Cohen subset of $\kappa $. On the other hand, in this model, $\kappa $'s supercompactness is indestructible under any $\kappa $-directed closed forcing which does not add a Cohen subset of $\kappa $.