The Generation of Closed Pointclasses

Assume $AD^++V=L(\mathcal P(\mathbb R))$. A pointclass $\Gamma\subsetneq\mathcal P(\mathbb R)$ is closed when $\mathcal P(\mathbb R)\cap L(\Gamma,\mathbb R)\subseteq\Gamma$. Let $w(\Gamma)=\sup\{w(A):A\in\Gamma\}$, where $w(A)$ is the Wadge rank, and let $\operatorname{Code}(\Sigma)$ denote a code for an iteration strategy. A Suslin cardinal is an ordinal with the source's standard determinacy-theoretic meaning.

Generation of closed pointclasses conjecture. If there is a Suslin cardinal $\kappa>w(\Gamma)$ and $L(\Gamma,\mathbb R)\vDash MC$, then for some hod pair $(\mathcal P,\Sigma)$,

$$w(\Gamma)\leq w(\operatorname{Code}(\Sigma)).$$

This conjecture is intended to produce a hod-pair strategy coding sets just beyond a closed pointclass, and is one of the three conjectural ingredients in the proposed proof of Mouse Capturing. Its notions are described by the source as somewhat informal.