This question is a follow up of this.

Is it consistent for *ALL* infinite stages $V_{\alpha > \omega}$ of the cumulative hierarchy of $\sf ZF$, to violate the weak partition principle? That is, each $V_{\alpha > \omega}$ can be parititioned into strictly more compartments than its elements.

The connection with the prior question is that if this question is answered to the positive, then the answer to the second question in the prior question would be answered to the positive as well.