Let $\kappa$ be supercompact in $V$. Let $\mathbb{P}$ be one of the standard forcing notions (or an iteration of such), and for simplicity assume that $\mathbb{P}$ is ${<}\kappa$-directed closed (e.g. a $\kappa$-Hechler iteration). In particular, $\kappa$ will remain supercompact in $V^{\mathbb{P}}$ if $V$ was 'Laver prepared'.
Question: What conditions can we impose on $\mathbb{P}$ and $V$ such that $\kappa$ remains supercompact in every intermediate forcing extension, i.e. $V \vDash \forall \mathbb{Q} \triangleleft \mathbb{P} \, \colon \, \Vdash_{\mathbb{Q}} \check{\kappa} \, \text{is supercompact}$ ?