Just because there is no evidence of a conspiracy by Burr doesn't mean one didn't actually happen. In fact, Conspiracy Theories and Secret Societies for Dummies (which is oddly a dubious source for the series) claims Wikinsson and Buur were workin for the Spanish
You can't in good faith use that argument though. That opens a doorway for every conspiracy theory and baseless claim out there. I could just as easily claim that the United States government perpetuated the September 11th Attacks and write about it being revealed in a timeline as if it is fact. Or defend that thread claiming awhile back that two random historical figures were in truth siblings, and made it simple fact in the world they were creating. I've nothing against religion, growing up with a type of flowing faith and religion myself, but I despise an unwavering literalism in every word of a religious text. Such an argument can be used to justify any thoughts within these books that can deny the world needed progress, or certain depths of curiousity.
... At its generalist, the assertion just because there isn't proof of something doesn't mean it isn't true, can be correct. It is in some cases for all we know, unlocked deep into the future by instruments far beyond our capabilities now We should always be inquiring of thoughts and ideas without dismissal; we've been incorrect so many times before. Still, it's dangerous to just allow that to be the titanium base on which any idea sits as reality.
I mean, in the case of Burr's trial specifically, it shows the unjust nature of such an idea. No proof of crime outside the biased account of the accuser. Pushed on, despite that lack, by Jefferson's political vendetta. Had he been found guilty, regardless of whether he actually plotted, it would be a travesty to just and fair trial.
... Now, sorry for being unbearably serious or over the top. I am interested to see what happens here. I only take issue with the use of that argument in debate, the mentioning of an admittedly dubious source besides

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Six words: Napoleon Bonaparte, Emperor of the French
Burr was no Napoleon though. Taking a note from the thread title, I'll express my concern and say I do hope you'll be true to Burr. Hamilton has a wonderful depiction of Burr, but a depiction undeniably biased to give Alexander Hamilton further light. Hamilton wasn't a saint, he was a man like any other. He was ruthless, charismatic, and had quite a lot of questionable ideas alongside the genius. I'm excited to see what you can do with this, there's a lot of stuff that can be done here. I do hope you won't just declare the exact plot he was accused of truth and have him immediately declare war against a nation he had part in building, regardless if he's viewed as anything more than Alexander's power hungry foil by pop culture. They were both human, with all the benefits and negatives entailed.
Oh, and two things lastly:
1.) I don't believe Burr supported Hamilton's suggestion of an elective monarchy. Not every republican will declare themselves emperor at first chance.
2.) He was a radical abolitionist.