The tree property at the ℵ2n's and the failure of SCH at ℵω

We show - starting from a hypermeasurable-type large cardinal assumption - that one can force a model where 2ℵω=ℵω+2, ℵω is a strong limit cardinal, and the tree property holds at all ℵ2n, for n>0. This provides a partial answer to the question whether the failure of SCH at ℵω is consistent with many cardinals below ℵω having the tree property.