  We ask when it is possible to construct arbitrary unitary multiplets of the
superconformal algebra with eight Poincar\'e supercharges that are compatible
with locality from (continuous deformations of) representations in free field
theories. We answer this question in two, three, and five dimensions. In four
dimensions, we resort to an intricate but self-consistent web of conjectures.
If correct, these conjectures imply various new non-perturbative constraints on
short multiplets in any local unitary 4d $\mathcal{N}=2$ superconformal field
theory and on an unusual set of related vertex algebras. Throughout, we connect
our results with properties of deformations in the space of theories.
