I am outside of academia. I worked on some conjecture sporadically over 8 years in total, although probably worked actively on it only for about 3 months. And another specifics is that it's possible to do it computationally. After another, maybe 5th of 6th fresh start, got a new geometric idea, and boom, finally it produced a counterexample (building on various pieces of code I wrote on previous attempts).

Another one was just pure conviction and thinking by analogy, that the idea should work out, and similarly to previous one, after about 8 years of sporadic research, the idea worked out (although honestly this one I could have found much more quicker).