The tree property at the double successor of a singular cardinal with a larger gap

Starting from a Laver-indestructible supercompact κ and a weakly compact λ above κ, we show there is a forcing extension where κ is a strong limit singular cardinal with cofinality ω, 2κ=κ+3=λ+, and the tree property holds at κ++=λ. Next we generalize this result to an arbitrary cardinal μ such that κ