The weak maximizing property for $(L_p([0,1]),L_q([0,1]))$

We solve [4, Question 4.2] left open by Dantas, Jung and Mart\'inez-Cervantes by providing a complete characterization for the pairs $(L_p([0,1]),L_q([0,1]))$ having the weak maximizing property. More precisely, we prove that, for $1<p<\infty$ and $1\leq q<\infty$, the pair $(L_p([0,1]),L_q([0,1]))$ has the weak maximizing property if and only if $p=q=2$. A key step in the proof is a Gaussian construction that yields the failure of the compact perturbation property for the pair $(\ell_2,L_r([0,1]))$ whenever $r>2$.