We argue that the observed UV finiteness of the 3-loop extended
supergravities may be a manifestation of a hidden local superconformal symmetry
of supergravity. We focus on the SU(2,2|4) dimensionless superconformal model.
In Poincare gauge where the compensators are fixed to phi^2= 6 M_P^2 this model
becomes a pure classical N=4 Einstein supergravity. We argue that in N=4 the
higher-derivative superconformal invariants like phi^{-4}W^2 \bar W^2 and the
consistent local anomaly delta (ln phi W^2) are not available. This conjecture
on hidden local N=4 superconformal symmetry of Poincare supergravity may be
supported by subsequent loop computations.Comment: 14 p A discussion of half-maximal D=6 superconformal models is adde
