Is there any way to have a scalar field that transforms non-trivially under local special conformal transformations? Just by the index structure, I can see that the possibilities are
$$\begin{align} \delta_K \phi &= \Lambda_{K}^\mu (x) \partial_\mu \phi \,, & \delta_K \phi &= x^\mu \Lambda_{K \mu} (x) \end{align}$$
where the subscript $K$ represents transformation with respect to the special conformal transformation, and $\Lambda_{K\mu} (x)$ is the parameter of special conformal transformations.
Considering the Poincare and dilatation transformations of the scalar $\phi$, I can see that both the first and the second possibilities are just orbital parts of Poincare and (inhomogeneous) dilatation transformations respectively.
Am I missing something? Is there really no way to write down a scalar that transforms non-trivially under local special conformal transformations?