= Solution
The scaling in part 5 gives
$$
\|u(t)\|_4^4=\lambda(t)^{-2}\|v\|_4^4
=\frac{\|v\|_4^4}{1+4t^2}.
$$
Thus $u\in L^4_{t,x}([-1,\infty)\times\mathbb R^2)$. For $T<T'$, the dual <Strichartz estimate for the free Schrödinger equation> gives
$$
\left\|\int_T^{T'}S(-s)(u|u|^2)(s)\,ds\right\|_2
\lesssim\||u|^2u\|_{L^{4/3}_{t,x}([T,T'])}
=\|u\|_{L^4_{t,x}([T,T'])}^3,
$$
which tends to zero as $T,T'\to\infty$. The integrals in the question are therefore Cauchy in $L^2$.
The <Duhamel principle> in the interaction representation defines an $L^2$ limit $u_\infty$ and expresses $S(-t)u(t)-u_\infty$ as the tail integral from $t$ to infinity. The same estimate sends that tail to zero, proving the <scattering from a finite Strichartz norm> conclusion
$$
\|u(t)-S(t)u_\infty\|_2\longrightarrow0.
$$
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