Assume we start with a ground model satifying GCH. What are some proper forcing notions with fusion (satisfying axiom A but not countably closed), which when iterated $\aleph_2$ times with countable support, preserve cardinals and make $\mathfrak{p}=\aleph_2$? Does such a forcing even exist?

For example, the prototypical such forcing, if in the above we replace $\mathfrak{p}$ with:

$\mathfrak{b}$, we get Laver forcing,

$\mathfrak{d}$, we get Miller forcing,

$\mathfrak{c}$, we get Sacks forcing, etc.

Thanks.

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