Saturation characterizations for suitable ultrafilter sequences

Let $T$ be a complete countable theory, let $\mathbf{s}$ be a suitable sequence, and let $\mathcal{U}_{\mathbf{s}}$ be the associated ultrafilter. Write $k_*(\mathbf{s})$ for the sequence parameter appearing in the amalgamation condition. Saturation characterization conjecture. The following equivalences hold: (I) if $\mathbf{s}$ is of type I, then $\mathcal{U}_{\mathbf{s}}$ $\lambda^+$-saturates $T$ if and only if $T$ is simple and has $\mathcal{P}^-(k_*(\mathbf{s}))$-amalgamation of models; (II) if $\mathbf{s}$ is of type II, then saturation is equivalent to $T$ being low and having this amalgamation property; (III) if $\mathbf{s}$ is of type III, then saturation is equivalent to $T$ being strongly low and having this amalgamation property; and (IV) if $\mathbf{s}$ is of type IV, then saturation is equivalent to $T$ being superlow and having this amalgamation property. These conjectured characterizations would unify the saturation criteria for the four types of suitable sequences, extending the known implications that type I and type II sequences characterize simplicity and lowness in the settings described earlier. The type III and IV notions are provisional in the paper, so those cases remain especially dependent on the proposed framework.