The Variety of Projection of a Tree-Prikry Forcing

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $\kappa$-distributive forcing notion of size $\kappa$ be extended to a $\kappa$-complete ultrafilter.