3 problems

Let $m<\omega$ and let $M$ be an $m$-sound, $(m,\omega_1+1)$-iterable premouse, and let $M$ be active, $1$-sound, and $(0,\omega_1+1)$-iterable in the second case. The questions ar…

Capturing of hod pairs conjecture. There is a $\Sigma$-iterate $\mathcal Q$ of $\mathcal P$ such that, if $\Lambda$ is the strategy of $\mathcal Q$ induced by $\Sigma$, then…

Iterability conjecture. Under these hypotheses, $M$ is $\omega_1+1$-iterable.