Iterated mantle and HOD coincidence conjecture

Let $V$ be a model of ${\rm ZFC}$ and let $\alpha\leq\mathop{\rm ORD}$. In another model $\overline{V}$ of ${\rm ZFC}$, write $({\mathord{\rm M}}^\alpha)^{\overline{V}}$ and $({\mathord{\rm gM}}^\alpha)^{\overline{V}}$ for the $\alpha$th inner mantle and inner generic mantle, and write $({\rm HOD}^\alpha)^{\overline{V}}$ and $({\mathord{\rm g}{\rm HOD}}^\alpha)^{\overline{V}}$ for the $\alpha$th iterated inner HOD and inner generic HOD. Iterated mantle and HOD coincidence conjecture. If $V$ is any model of ${\rm ZFC}$, then for any $\alpha\leq\mathop{\rm ORD}$, there is another model $\overline{V}$ of ${\rm ZFC}$ in which $V$ is simultaneously the $\alpha$th inner mantle, the $\alpha$th generic inner mantle, the $\alpha$th inner HOD, and the $\alpha$th inner generic HOD:

$$V=({\mathord{\rm M}}^\alpha)^{\overline{V}}=({\mathord{\rm gM}}^\alpha)^{\overline{V}}=({\rm HOD}^\alpha)^{\overline{V}}=({\mathord{\rm g}{\rm HOD}}^\alpha)^{\overline{V}}.$$

This extends the preceding inner-mantle conjecture to generic mantles and iterated HOD constructions, asserting that all four processes can be arranged to reach any prescribed model of ${\rm ZFC}$ at any ordinal stage; its status is open.