Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular representation?
If this is true, then we would have a Gaussian action which is free and ergodic and not weakly contained (as an action) in the Bernoulli action. So, the second question:
Are there countable Property (T) groups (or more generally, non-amenable groups), that have a free and ergodic Gaussian action not weakly contained in the Bernoulli action?