On patterns of cardinals with the tree property

Building upon work of Abraham (cf. [1]), Cummings and Foreman have shown in [2], starting from ω many supercompact cardinals, that consistently the following holds. (⋆) For every n < ω, 2n = אn+2 and אn+1 has the tree property. Recall that a cardinal κ is said to have the tree property if there is no Aronszajn κ-tree, i.e. if every tree of height κ all of whose levels have size < κ admits a cofinal branch. We here show (in a certain sense of ”show”):