  Using the superconformal framework, we construct a new off-shell model for
$\cal N=4$ topologically massive supergravity which is minimal in the sense
that it makes use of a single compensating vector multiplet and involves no
free parameter. As such, it provides a counterexample to the common lore that
two compensating multiplets are required within the conformal approach to
supergravity with eight supercharges in diverse dimensions. This theory is an
off-shell $\cal N=4$ supersymmetric extension of chiral gravity. All of its
solutions correspond to non-conformally flat superspaces. Its maximally
supersymmetric solutions include the so-called critical (4,0) anti-de Sitter
superspace introduced in arXiv:1205.4622, and well as warped critical (4,0)
anti-de Sitter superspaces. We also propose a dual formulation for the theory
in which the vector multiplet is replaced with an off-shell hypermultiplet.
Upon elimination of the auxiliary fields belonging to the hypermultiplet and
imposing certain gauge conditions, the dual action reduces to the one
introduced in arXiv:1605.00103.
