Do you believe that Fyodor Kuzmich was really Alexander I or that it was just a false rumor?

Well, I like me some conspiracy theories, but as much as I want to believe, I'm gonna say it's just a false rumor.

The guy was about Alexander's age, spoke several foreign languages(uncommon, especially in Tomsk) and refused to speak about his past. And Alexander did not like being Tsar, as he felt that he was responsible for his fathers death(coup) and the deaths of the soldiers in the Napoleonic wars.

Are there any primary source documents that speak to this rumor or is the myth/rumor based on word of mouth only?

Also Alexander II visited him and Nicholas II visited his grave.

remember the case with the Nicolas II daughter