Weaken the hyperhuge assumption in the bedrock existence result

Investigate whether the conclusion that a bedrock model exists and is distinct from V, with κ hyperhuge in the bedrock, can be derived from the existence of a tightly P-Laver-generically large cardinal of strictly lower consistency strength than hyperhuge for nice iterable classes P of posets.

Background

Proposition 2.13 shows that for a nice iterable class P of posets, the existence of a tightly P-Laver-generically hyperhuge cardinal forces the bedrock to exist and differ from V, thereby implying ¬GA.

The authors note that related results establish consequents under ultrahuge assumptions but do not know whether the hyperhuge hypothesis can be weakened to a strictly lower large-cardinal strength while preserving the same conclusion.