I have been interested in the attempts to prove this Conjecture since 2000 and like many others (eg Ken Conrow) I have tried to find a convincing solution. Today I read on this forum what looks like a proof that there cannot be an internal cycle beyond 4:2:1 but I don't think the author realizes it as that. Of course that still doesn't "prove" the whole conjecture because it does not address an infinitely long trajectory. I wonder if anyone else has done so. I have created a system to account for every integer possible and its first ascent step that may answer the density problem in nearly every proof offered so far.