Classification of multi-solitons in one sense of time for the 5D energy-critical wave equation

We classify the multi-solitons, in the positive sense of time, of the focusing energy-critical wave equation in space dimension $5$, more precisely the solutions $u(t,x)$ of the equation $${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u-\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{3}}u=0,(t,x)\in [T_0,\mathrm{\infty })\times {\mathbb{R}}^5,$$ which satisfy the estimate $${\parallel {\mathrm{\nabla }}_{t,x}(u(t)-\sum _k{W}_k(t))\parallel }_{L^2}\lesssim t^{-{\frac{1}{2}}^{+}}\text{for all }t\gg 1\mathrm{.}$$ Here, $\{W_k{\}}_k$ is any given finite family of traveling waves, based on the explicit ground state solution $$W(x)={(1+\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{15})}^{-\frac{3}{2}}$$ with collinear two-by-two different velocities.