  This paper is the the third part of a series of paper whose aim is to use of
the framework of \emph{twisted spectral triples} to study conformal geometry
from a noncommutive geometric viewpoint. In this paper we reformulate the
inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral
triples. This involves a notion of Poincar\'e duality for twisted spectral
triples. Our main results have various consequences. In particular, we obtain a
version in conformal geometry of the original inequality of Vafa-Witten, in the
sense of an explicit control of the Vafa-Witten bound under conformal changes
of metric. This result has several noncommutative manifestations for conformal
deformations of ordinary spectral triples, spectral triples associated to
conformal weights on noncommutative tori, and spectral triples associated to
duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes
conjecture.
