  We prove a local index formula for a class of twisted spectral triples of
type III modeled on the transverse geometry of conformal foliations with
locally constant transverse conformal factor. Compared with the earlier proof
of the untwisted case, the novel aspect resides in the fact that the twisted
analogues of the JLO entire cocycle and of its retraction are no longer
cocycles in their respective Connes bicomplexes. We show however that the
passage to the infinite temperature limit, respectively the integration along
the full temperature range against the Haar measure of the positive half-line,
has the remarkable effect of curing in both cases the deviations from the
cocycle identity.
