The saturation of club guessing ideals

We prove that it is consistent that there exists a saturated tail club guessing ideal on ω1 which is not a restriction of the non-stationary ideal. Two proofs are presented. The first one uses a new forcing axiom whose consistency can be proved from a supercompact cardinal. The resulting model can satisfy either CH or 2ℵ0=ℵ2. The second one is a direct proof from a Woodin cardinal, which gives a witnessing model with CH.