Let's say that someone lied about jury nullification in order to get on a jury with the intent to nullify, and they then get indicted for perjury. If they just demanded a trial by jury, would the facts of their case require that their jury learn about jury nullification? If so, would all they have to do is convince them that this whole case is absurd, in order to be found not guilty of nullification due to their own jury nullifying the charges?
The theory of jury nullification would likely be relevant to the case, and the defendant would likely be entitled to give the jury at least some explanation of what it is and how it works.
I think you could expect the court to take extra measures to ensure that the jury was screened for openness to nullification, and to really hammer on that message in the jury instructions, which would likely provide additional detail to the jury about how and why nullification is illegal.
If the defendant could, despite all that, convince the jury that the case is absurd, nullification would be just as much an option in the second case as in the first.
I very much doubt that a perjury case would be brought against a juror in these circumstances.
Perjury cases resulting from court trials are vanishingly rare, and perjury cases resulting from a juror's statement about their intentions rather than about a specific presently existing fact are even more rare.
Proof of the case would almost surely rest on evidence regarding what actually happened in the deliberations of the jury and that would be inadmissible evidence because they are deliberations (absent a motion to set aside a verdict due to racial discrimination). So, there would really be no way to prove that case.
It might very well be dismissed in a preliminary hearing or in motion practice prior to trial.