Soliton resolution for equivariant wave maps from hyperbolic space

Consider the equivariant wave-map Cauchy problem on $\mathbb{R}\times\mathbb{H}^2$ with finite energy initial data $(\psi_0,\psi_1)$. The finite energy harmonic maps are

$$P_\lambda(r)=2\operatorname{arctanh}\bigl(\lambda\operatorname{arctanh}(r/2)\bigr),\qquad \lambda\in[0,1),$$

and define

$$\lambda=\tanh\bigl(\psi_0(\infty)/2\bigr)\in[0,1).$$

Soliton resolution conjecture. The Cauchy problem is globally well-posed, and its solution scatters to $P_\lambda$ as $t\to\pm\infty$. The harmonic maps $P_\lambda$ minimize the energy in the corresponding endpoint classes, and the source notes that the conjecture was verified for endpoints $\lambda\leq\Lambda_0$ with $\Lambda_0\geq0.57$, leaving $\lambda\in[\Lambda_0,1)$ open.