  We build an example of a non-transitive, dynamically coherent partially
hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth
in its fundamental group such that $f^n$ is not isotopic to the identity for
all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main
idea is to consider a well-understood time-$t$ map of a non-transitive Anosov
flow and then carefully compose with a Dehn twist.
