You would think that his lying was fact, not contrary to fact, so you don't use the subjunctive in that part of the sentence.
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SJ88To use "were" in that case sounds awkward and wrongIt is awkward and wrong. There is no reason why "think that" or "thought that" should ever be followed by a subjunctive. Ignore the subjunctive in the if clause. That's got nothing to do with the choice in the clause you're puzzled about.