Each option is I think a legitimate interpretation - and context will
need to determine between them (and maybe even then it will be
ambiguous).
On example of each where I think it is clear:
Acts 3:16 where "his name" I think is fairly clearly an objective
genitive.
Rom 3:3 where pistin tou theou I think needs to be understood as the
faithfulness of God.
Rom 3:22 is also an example like Gal 2:20 where I have read and heard
both main options argued.
Perry Wiles