First of all, there already was a conspiracy--Lincoln wasn't the only victim either. The Secy. of State Seward was knifed but managed to survive the attack, and a third man was supposed to kill Andrew Johnson but he either didn't find him or he abandoned the mission.
The Problem is that Evidence really started coming like that there would be a difficult political decision made, one similar to the Death of John F. Kennedy--where a special commission is made and the outcome altered for political reasons. The US government can not afford to go on a purge of a northern conspiracy--Indeed, the CSA army under Longstreet is still around, and it won't surrender for another two days, to say nothing of CS Partisans.
The outcome of the convention would probably rule that John Wilkes Booth and his pals acted on their own initiative. It would be a dirty secret, but the people in power are OBVIOUSLY NOT going to allow their guilt to be discovered. Indeed, because Booth was the triggerman, blame is instead going to be focused on the Confederates, many of whom celebrated the death of Lincoln.
This Authentic Evidence is destroyed or deleted--it would break the country--and the lack of a Federal police at the time means that the Pinkerton Mercenaries are probably called in to secure the information. There would be ominous rumors, but the commission would be accepted and there would just be a JFK myth for Lincoln's death.
Besides, the involvement of this many important people is too lunatic to be believed. In a way, the sheer silliness of this happening serves as a mental barrier from figuring it out.