This is a situation in which you are never going to win, so if this shortens the pain, then it has to be considered.
If you know the petition is bogus, then that is all that matters.
No-one is going to be waiting at the end of this with a gold medal because you prolonged your pain for the sake of proving a point. And to whom? You know what the situation was, and you can live with your head held high afterwards.
The opinions of other people really don't matter - tell the truth to those you care about, let the rest think what they will.
Hope this helps.