Let $E$ be a Banach lattice. A subset $A$ of $E$ is called solid if $|x|\leq |y|$ for some $y\in A$ implies that $x\in A$. For a subset $A$ of $E$, the solid hull of $A$ is the smallest solid set including $A$ and is exactly the set $$\operatorname{Sol}(A):=\{x\in E:\exists y\in A \operatorname{with} |x|\leq |y|\}.$$
Question. Under what hypothesis on a Banach lattice $E$, the following inclusion $$\overline{\operatorname{Sol}(A)}^{\sigma(E^{**},E^{*})}\subseteq \operatorname{Sol}(\overline{A}^{\sigma(E^{**},E^{*})})$$ holds for each norm bounded subset $A$ of $E$ ?
