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Sorry but I would like to change my vote and would now argue for a positive answer. I am afraid I made a mistake when saying that I know a distribution on $sl(2)$ supported on the nilcone whose FT also has this property. I know one for $sl(2)$ over a finite field but not over a local non-Archimedian field. In this latter context it is known (at least when characteristic of the local field is zero) that FT of a nilpotent orbit comes from an $L^1$ function on the set of regular semi-simple elements, this shows that FT of a nilpotently supported invariant distribution can not be nilpotently supported. Another argument that might show the same without assuming the distribution is invariant is like this: the metaplectic group Mp(2) acts on the space of distributions where upper triangular matrices with 1 on diagonal act by multiplication by an additive character of the quadratic form and the order 4 element acts by FT. If both $\psi$ and FT of $\psi$ are supported on the nilcone, then $\psi$ is invariant under this action. Then perhaps the center of Mp(2) acts by a nontrivial character, so no nonzero distribution is invariant under Mp(2)?