The tree property below ℵω⋅2ℵω⋅2

We improve the best known result on successive regular cardinals with the tree property. In particular we prove that relative to an increasing ω+ωω+ω-sequence of supercompact cardinals it is consistent that every regular cardinal on the interval [ℵ2,ℵω⋅2)[ℵ2,ℵω⋅2) has the tree property.