  The work of Mann and Rafi gives a classification surfaces $\Sigma$ when
$\textrm{Map}(\Sigma)$ is globally CB, locally CB, and CB generated under the
technical assumption of tameness. In this article, we restrict our study to the
pure mapping class group and give a complete classification without additional
assumptions. In stark contrast with the rich class of examples of Mann--Rafi,
we prove that $\textrm{PMap}(\Sigma)$ is globally CB if and only if $\Sigma$ is
the Loch Ness monster surface, and locally CB or CB generated if and only if
$\Sigma$ has finitely many ends and is not a Loch Ness monster surface with
(nonzero) punctures.
