No matter what they say about him, it's false.
Can you have a comma here.
There feels like a pause after him but you're breaking the sentence up with the comma. Perhaps dash would be better or no punctuation.
Yes. In fact, a comma belongs there. " The rest is a long introductory phrase, which is generally followed by a comma, as you have done.
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Yes. In fact, a comma belongs there. The essence of the statement is "it's false." The rest is a long introductory phrase, which is generally followed by a comma, as you have done.