I have 4 parrots with my 2 cats, one is an F4 savannah. I have an Amazon, African Grey, Mini Macaw and a Pionus....the pionus being the smallest of the 4, but still bigger than your green cheek conure.
3 of my parrots (macaw, pionus and Grey) live in a 'parrot room' (mainly so we can shut the door when they get crazy noisy), other than that, the door is always open and there are litter boxes in there since Tiger (dsh) loves my parrots. Our Amazon doesn't care to associate with a flock and is in our living room.
When I first brought Zeddie home (F4), after her 2 weeks quarantine, I harnessed her and took the birds out. I let her wander around them to get acquainted, and I let them open their beaks at her, so she could understand they can hurt her. Anytime she showed interest in them as prey, she has been tossed out of the room. She has climbed on their cages before....because she was curious about their food. She climbed into my Grey's cage for the same reason. My Grey just looked at her and yelled at Tiger (I mean, they are both cats so she was close).
I think a lot of it depends on introductions. Prepare for the worst but hope for the best. Also, be careful how you treat your bird. Savannahs are so bonded to their owners, you wouldn't want your savannah to think that the bird is a nuisance. If Zeddie does feel the need to stalk a parrot, she will ALWAYS stalk our amazon because she is the noisiest and drives more people crazy. That being said, my Amazon's cage is far off the ground and has a lot of space to move around in.
Our parrots do come out with Zeddie. She usually just stays away then...unless my macaw is running around, then she watches his tail since it is very long.
I also never use feathers on cat toys, and I never buy birdie cat toys.
Their compatibility would vary on generation of savannah, as well as each individual personality.