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, $2^{\kappa }={\kappa }^{+3}={\lambda }^{+}$, and the tree property holds at ${\kappa }^{++}=\lambda$. Next we generalize this result to an arbitrary cardinal μ such that $\kappa <\mathrm{cf}(\mu )$ and ${\lambda }^{+}\le \mu$. This result provides more information about possible relationships between the tree property and the continuum function.