Supercompact Laver-genericity versus the Ground Axiom

Determine whether the existential axiom asserting the existence of a tightly P-Laver-generically supercompact cardinal implies the negation of the Ground Axiom for natural iterable classes P of posets (for example, proper, semiproper, or stationary preserving posets).

Background

The authors separate various restricted Maximality Principles from Recurrence Axioms using GA and show that some stronger Laver-generic cardinal principles lead to ¬GA. Establishing that tightly P-Laver-generically supercompact implies ¬GA would provide a clear separation between Laver-genericity and certain double-plus forcing axioms.

This problem aims to clarify the exact strength of supercompact Laver-genericity relative to GA, thereby refining the landscape of implications among forcing axioms, recurrence/maximality principles, and Laver-generic large cardinals.