$C^r$-Chain closing lemma for certain partially hyperbolic diffeomorphisms

For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with 1-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism $f$, if a point $y$ is chain attainable from $x$ through pseudo-orbits, then for any neighborhood $U$ of $x$ and any neighborhood $V$ of $y$, there exist true orbits from $U$ to $V$ by arbitrarily $C^r$-small perturbations. As a consequence, we prove that for $C^r$-generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.