Global asymptotic stability of explicit hyperbolic solitons

Let $(u_0,u_1)$ be smooth finite energy initial data for wave maps

$$(u_0,u_1):\mathbb{H}^2\to\mathbb{H}^2$$

with spacetime domain $\mathbb{R}\times\mathbb{H}^2$. Fix $\lambda\in[0,1)$ and suppose that, for some compact set $K\Subset\mathbb{H}^2$,

$$u_0(x)=\mathcal{P}_\lambda(x)\qquad\text{for }x\in\mathbb{H}^2\setminus K.$$

Non-equivariant soliton resolution conjecture. The unique wave-map evolution associated with $(u_0,u_1)$ is globally regular and scatters to $(\mathcal{P}_\lambda,0)$ as $t\to\pm\infty$. This is presented as a more ambitious nonequivariant form of soliton resolution; the source gives no resolution, while emphasizing that it would imply global asymptotic stability under arbitrarily large non-equivariant perturbations.