Let's call a cardinal $\delta$ an $\text{I1}$-tower cardinal if for each $A\subseteq V_{\delta}$, there exists a $\kappa<\delta$ such that whenever $\kappa<\alpha<\delta$ there is some $\lambda<\delta$ and a $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and $j(\kappa)>\alpha$ and $j(V_{\lambda}\cap A)=V_{\lambda}\cap A$.

Is the existence of an $\text{I1}$-tower cardinal consistent? Is the existence of an $\text{I1}$-cardinal implied by the existence of an $\text{I0}$-cardinal? Does $\text{Con(I0)}$ imply $\text{Con}(\text{there is an I1-tower cardinal})$? Does $\text{Con}(\text{there is an I1-tower cardinal})$ imply $\text{Con(I0)}$?