In the introduction of this paper at the top of p11, it is mentioned that a hypothetical enhancement of the scale invariance of the CMB spectrum to conformal invariance could potentially be established by analyzing the higher n-point functions of the tensor modes.

First of all, why are the tensor modes needed for this, why can conformal invariance not be seen in the scalar modes too, if it is there?

How does this work exactly (mathematically), how does conformal invariance (not just scale invariance...!) manifest itself in the n-point functions of the tensor modes?

Has this already been looked at or will it be done for the tensor modes in the BICEP2 data for example?