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My impression is that Johannes asks whether it is consistently possible to arrange that the bound on the size of the length of the sad's is strictly less than the saturation of NS.
I should probably add that the motivation for this question has nothing to do with questions concerning cardinal characteristics, so I am not worried about $\aleph_2$ being collapsed. An idea how to solve the problem under CH would be helpful as well.
Yes, I recall learning that from one your answers here. I picked $Add(\omega, \omega_1)$ as it is closer to the actual problem I am facing. Plus it rules out some tricks involving rearranging of the factors.
Have not thought about it at all. But Lyubomyr Zdomskyy just mentioned to me that in models of $\mathsf{PFA}(S)$, where $S$ is a Suslin tree, after forcing with $S$ the $PID$ holds, hence there are no Suslin trees anymore, so the opposite is consistent as well.
