I only know of this problem being discussed for scalars. In AdS, there's a unique SO(d-1,2) invariant vacuum, so your question doesn't apply. In de Sitter space, on the other hand, you have a one-parameter family of dS invariant vacua, labeled by a complex parameter alpha. Switching between these vacua can be accomplished by what's called a Mottola-Allen transform, and this corresponds to perturbing the CFT by some marginal deformation (at least in three-dimensional dS). See Bousso, Maloney, and Strominger for details.
I'm not really sure if these alpha vacua are so physical though. Taking the standard Euclidean vacuum, which is the analytic continuation of the vacuum from the sphere, corresponds to demanding that the fields start as plane waves, which sounds pretty reasonable. Also, Harlow and Stanford show that analytically continuing the infrared wavefunction from AdS gives the dS wavefunction with Euclidean initial conditions, so the alpha vacua are in some sense not as "preferred".
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