Rank cutoff at the first inaccessible limit of inaccessibles ($V_{\iota }\models \mathrm{ZFC}+\infty \mathrm{IC}+\neg \mathrm{ILI}$)

If $\iota$ is the least inaccessible limit of inaccessibles, $V_{\iota }$ models ZFC and internally has unboundedly many inaccessible cardinals, while none of its ordinals is an inaccessible limit of inaccessibles. Rank agreement makes inaccessibility below $\iota$ absolute. If a model of unboundedly many inaccessibles has no such limit in the first place, no cutoff is needed. This proves the corresponding relative-consistency nonimplication.