(BSG) Increasing Paranoia with Executions

I posted this over at BoardGameGeek, and unsurprisingly it wasn't a very popular idea.

So I'm thinking,
When I first heard about Exodus, and played my first couple of games with it, I liked the new rules for the Loyalty Deck partially because a character wasn't guaranteed to be Human when coming into the game to replace an Executed character.

But in practice, the newly-drawn card after being Executed is rarely "You Are A Cylon"; it might be a Final Five or a Personal Goal, but those don't really hurt the team that bad.

So here is my idea:

At game setup, create the Loyalty Deck as instructed by the Exodus rulebook. Follow the rules as-printed until the sleeper phase.

After the Sleeper Phase, the first time a character is executed, instead of shuffling a new You Are Not A Cylon card into the deck before drawing your new loyalty, shuffle a "You are a Late Cylon" card in.
(This could use a better name, but I don't even know if the mechanic's any good)

YOU ARE A CYLON

Action: Reveal this card.
You may not use the "Cylon Fleet" or "Basestar Bridge" location.

Only do this once. Future executions are handled by the normal rules.

It makes you a worse Cylon than the Sympathizer, for sure. It barely gives you an incentive to Reveal. But it adds more uncertainty to the Airlock As Cylon Detector, and maybe people will start using the Brig more often again?

What do you think?