Does there exist a complete and decidable axiomatization of the modal logic governing the forcing relation over all models of ZFC simultaneously, rather than over any fixed ground model?

The internal modal logic of forcing assigns modal operators to set-theoretic statements by interpreting possibility as truth in some forcing extension and necessity as truth in all forcing extensions. Hamkins and Lowe characterized this logic as S4.2 when working over a fixed ground model, but the question of whether a single clean modal logic captures the forcing relation in a way that is uniform across all possible ground models of ZFC remains unresolved. The challenge is to determine whether the collection of modal validities that hold simultaneously in every model of ZFC, regardless of which model you start from, forms a recursively axiomatizable and semantically complete system, or whether the global behavior of forcing introduces irreducible complexity that escapes any finitely describable modal framework.