I can't speak for them, but from what is reported there doesn't appear to be the same belligerent insistence that they believe she could still be alive. I agree with the rest of your post. It doesn't satisfy me either. Absolute proof isn't easy in many cases unfortunately, though.
I would not have used 'belligerent' insistence to describe the fight Kate and Gerry have had to have Madeleine's disappearance properly investigated by the police.
belligerent adjective
hostile and aggressive.
"the mood at the meeting was belligerent"
The hostility and aggression was directed against them sometimes even on this forum.
In the main the information circulating regarding this horror Christian Brueckner, has left me feeling incredibly deflated and I do not know any of the victims involved.
The hell that Kate and Gerry and all Madeleine's family are going through must be indescribable ~ and at the moment the words you have chosen are in my opinion inappropriate.