Guessing models and the approachability ideal

Starting with two supercompact cardinals we produce a generic extension of the universe in which the principles ${\rm ISP}(\omega_2)$ and ${\rm ISP}(\omega_3)$ hold simultaneously, and the restriction of the approachability ideal $I[\omega_2]$ to the set of ordinals of cofinality $\omega_1$ is the non stationary ideal on this set.

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