Even AI uses putative. Because there is zero ways to confirm biological paternity at this remove in time. The fact that Ser Piero acknowledged him and provided for him is indicative that he thought he was the father, but it’s not proof.
Those DNA tests should make the situation more clear, although I think the woman would not necessarily had much freedom of movement or freedom from gossip if she was cheating on a socially powerful lover. Also if she had a bad reputation the grandfather probably would not have taken the tax responsibility. Remember these societies lived by family connections and knowing what was happening with whom.