Strengthened disjunction or non-classical conjunction?

Haslinger and Schmitt examine the strengthened and non-classical conjunction. It has been claimed that the coordinators manu in Warlpiri or in child English and ya in Japanese lexically express disjunction, but are obligatorily strengthened to a conjunctive meaning in upward-monotonic contexts by some version of double exhaustification. Data with non-distributive predicates would thus be crucial to decide in supporting a strengthening mechanism deriving a conjunctive meaning for disjunction, or reflect independently attested non-classical properties of natural language conjunction.