  A pair of Banach spaces $(E, F)$ is said to have the weak maximizing property
(WMP, for short) if for every bounded linear operator $T$ from $E$ into $F$,
the existence of a non-weakly null maximizing sequence for $T$ implies that $T$
attains its norm. This property was recently introduced in an article by R.
Aron, D. Garc\'ia, D. Pelegrino and E. Teixeira, raising several open
questions. The aim of the present paper is to contribute to the better
knowledge of the WMP and its limitations. Namely, we provide sufficient
conditions for a pair of Banach spaces to fail the WMP and study the behaviour
of this property with respect to quotients, subspaces, and direct sums, which
open the gate to present several consequences. For instance, we deal with pairs
of the form $(L_p[0,1], L_q[0,1])$, proving that these pairs fail the WMP
whenever $p>2$ or $q<2$. We also show that, under certain conditions on $E$,
the assumption that $(E, F)$ has the WMP for every Banach space $F$ implies
that $E$ must be finite dimensional. On the other hand, we show that $(E, F)$
has the WMP for every reflexive space $E$ if and only if $F$ has the Schur
property. We also give a complete characterization for the pairs $(\ell_s
\oplus_p \ell_p, \ell_s \oplus_q \ell_q)$ to have the WMP by calculating the
moduli of asymptotic uniform convexity of $\ell_s \oplus_p \ell_p$ and of
asymptotic uniform smoothness of $\ell_s \oplus_q \ell_q$ when $1 < p \leq s
\leq q < \infty$. We conclude the paper by discussing some variants of the WMP
and presenting a list of open problems on the topic of the paper.
