Consistency of tightly P-Laver-generically huge cardinals with the Ground Axiom

Determine whether the existence of a tightly P-Laver-generically huge cardinal is inconsistent with the Ground Axiom for iterable classes P of posets; equivalently, show whether GA must fail assuming there exists a tightly P-Laver-generically huge cardinal.

Background

The paper proves that several stronger Laver-generic large cardinal assumptions, such as tightly super C(∞) P-Laver-generically ultrahuge and hyperhuge, imply the failure of GA in many natural instances of P.

However, for the (tightly) P-Laver-generically huge cardinal, the authors do not know whether GA must fail, leaving the relationship between this Laver-genericity level and the Ground Axiom unresolved.