I'm not clear on what exactly you're asking, just because of the phrasing of the last couple of paragraphs. However, it's certainly permissable to place a piece of the conspiracy that connects back to the central piece, but rams it's only line(s) straight into the edge of the puzzle, or onto the blank side of another piece. The only rule with placement it that it has to connect one edge of the piece that has a line to one edge of a piece alread in play with a line. It's actually a fairly effective strategy to make sure the conspiracy doesn't always connect to everything--for instance, if you've a character with a lot of Corp favors, you certainly want it to connect with that, but if you have no Society favors and one of the others has a half-dozen, it's best to run the line going that direction into a wall before it can get there. Same if your guilty hunch just got killed--why bother letting someone else's guilty hunch pay off an extra 5 VP?
As for is it possible to keep the conspiracy from connecting with anything, it's certainly possible within the rules, but it would take a concerted effort by multiple players to do so. The conspiracy is too vast and complex for a single person to completely nerf it.