I think we are forgetting the main point if we are talking about lying without considering its effect.
A white lie which doesn't actually effect anyone in comparison to a lie that may increase a person's trade risk or trustworthiness would be very different to situations to one another.
That wasn't the main point here because you didn't present your initial question in that manner.
So, to address your question once more:
"If someone is a proven liar, should they get negative trust?"Short answer: No.
Long answer: Please refer to my initial reply in this thread.
<...> it's kind of a tricky thing that can change depending on the situation. Not all lies are the same, if you get my drift.