Consistency of failure of Gorenstein projective precovering/deconstructibility

Determine whether it is consistent with ZFC that there exists a ring R such that the class 𝔾ℙ_R of Gorenstein projective modules is not precovering or not deconstructible.

Background

While large-cardinal hypotheses imply that 𝔾ℙ_R is deconstructible over every ring, the paper highlights the absence of a ZFC determination of the consistency status of the opposite scenario: a ring where Gorenstein projectives fail to be precovering or deconstructible.

Establishing (in)consistency would quantify the set-theoretic strength needed to ensure robust approximation properties in Gorenstein projective theory across all rings.