The common-iterate conjecture for generic generators

Let $V$ satisfy $obreak\textsf{AD}_{\mathbb R}+ obreak\textsf{NLE}+V=L({\wp}(\mathbb R))$. Let $obreak\Delta\subseteq{\wp}(\mathbb R)$ and let $g\subseteq\operatorname{Coll}(\omega_1,\mathbb R)$. In $V[g]$, let $(\mathcal P,\Sigma)$ and $(\mathcal Q,\Lambda)$ be generic generators for $obreak\Delta$ of the same type, either both pure extender pairs or both hod pairs, with $\sup(W(\mathcal P)\cup W(\mathcal Q))<\omega_1$. Common-iterate conjecture. In $V[g]$, there is a common iterate $(\mathcal R,\Psi)$ of $(\mathcal P,\Sigma)$ and $(\mathcal Q,\Lambda)$ such that, for the iteration embeddings $i:L[\mathcal P]\to L[\mathcal R]$ and $j:L[\mathcal Q]\to L[\mathcal R]$, one has $i(\omega_1)=\omega_1=j(\omega_1)$. The text states that this comparison result would suffice, together with the preceding non-badness conjecture, to establish LEC or HPC; it is proved only under additional hypotheses such as NWLW.