I'm not quite sure what you mean by the ordering by Γ-soundness (for the various Γ you mention).  Of course they're all sound, because they're all true.  Maybe you mean the collection of their Γ consequences?  I would expect that order to be the same.  Sure, V=L has  consequences that are contradicted by (say) the existence of a measurable, but V=L is not a large-cardinal hypothesis.