Countable-to-arbitrary coproduct closure for Gorenstein injective modules

Determine whether, for a ring R, closure of the class of Gorenstein injective R-modules under countable coproducts implies closure under arbitrary coproducts.

Background

Theorem A establishes that, for a locally finitely generated Grothendieck category, closure of the Gorenstein injective objects under arbitrary coproducts is equivalent to the Gorenstein injective class being covering and to several other conditions. The authors isolate the weaker hypothesis of closure under countable coproducts and ask whether it already forces closure under coproducts of every cardinality for module categories.

The paper notes that an affirmative answer has been obtained under the set-theoretic assumption V=L. Consequently, assuming the consistency of ZFC, a counterexample cannot be constructed in ZFC alone; the question remains unresolved in the stated generality.