  Quite recently, a new property related to norm-attaining operators has been
introduced: the weak maximizing property (WMP). In this note, we define a
generalised version of it considering other topologies than the weak one
(mainly the weak$^*$ topology). We provide new sufficient conditions, based on
the moduli of asymptotic uniform smoothness and convexity, which imply that a
pair $(X,Y)$ enjoys a certain maximizing property. This approach not only
allows us to (re)obtain as a direct consequence that the pair $(\ell_p,\ell_q)$
has the WMP, but also provides many more natural examples of pairs having a
given maximizing property.
