We know that there are examples of Cox rings of Mori Dream Spaces that are not Cohen-Macaulay, for instance, the Cox ring of a very general algebraic hyper-Kähler 4-fold. But, this Cox ring seems to be Du Bois (please, correct me if this is wrong).
i) Do we know examples of Mori Dream Spaces that are not Du Bois? (I expect the answer to be yes, but I don't have an example).
ii) Do we know examples of Cox rings of Mori Dream Spaces that are not Du Bois?