  In this paper, we study the critical norm conjecture for the inter-critical
nonlinear Schr{\"o}dinger equation with critical index $s_c$ satisfying
$\frac{1}{2}<s_c<1$ when $d\geq 5$. Under the assumption of uniform boundedness
of the critical norm, we prove the global well-posedness and scattering for the
Cauchy problem. We follow the standard `Concentration compactness/Rigidity
method' established in \cite{KenigMerle1,KenigMerle2}, and treat three
scenarios for the critical element respectively. Moreover, double Duhamel
method and interaction Morawetz estimate are applied to exclude the critical
element.
