The paper on strong completeness of the logic J establishes that derivability from an arbitrary set of premises matches semantic consequence, but the boundary conditions of this result leave open a precise question: for which classes of models and which cardinalities of premise sets does strong completeness hold for intuitionistic and related intermediate logics beyond J? Specifically, it is unclear whether the techniques used generalize to uncountable premise sets without imposing compactness-style restrictions, or whether there exist natural extensions of J for which strong completeness fails at uncountable cardinality while remaining valid at countable cardinality. This gap between the finite, countable, and uncountable regimes is not addressed by current methods.