Ramza:
The one Glaring Flaw The Ramza sees is if Fran is not a Gun owner, then that means The Ramza has to figure out who the Prostitute is.
If The Ramza Can't we learn two things, that person is not the Mafia Prostitute and Rinoa is a liar who we need to lynch the next day.
Also, if Edgar is a cop, then he is straight. As I came up innocent.
That would mean if Fran is a cop, she's insane.
In which case Fran, The Ramza suggests that in order to help the town if you are a cop to post your results, that way if you are in fact Insane then we know what you learned in your investigations and we can use it.
If you are a regular cop, we can lynch Edgar next, as he's obviously lying.
Edit: The Ramza also suggests you vote now, because if Fran doesn't come back in time, you won't have a vote and will be voting for yourself.
Edit2: Also Rinoa, a death of Fran will confirm your status in this game. It will at least clear you as innocent if Fran is Lynched and has a role with a gun.
If its the real cop, we can kill Edgar for lying. If its an insane cop, we can learn of more people who are innocent or not.
If its a paranoid gun order, we can stop potential town deaths.
MAfia or not, Fran is good choice for a lynching.