Global regularity conjecture for wave maps

Let ${\mathcal S}$ be the space of classical data $(\phi_0,\phi_1)$, where $\phi_0:{\mathbf R}^2\to{\mathbf H}$ is Schwartz modulo constants and $\phi_1:{\mathbf R}^2\to T{\mathbf H}$ is Schwartz with $\phi_1(x)\in T_{\phi_0(x)}{\mathbf H}$ for every $x$. A classical wave map is a map $\phi:{\mathbf R}\times{\mathbf R}^2\to{\mathbf H}$ solving the wave-map equation, with initial data $\phi[0]=(\phi_0,\phi_1)$. Global regularity for wave maps. For every $(\phi_0,\phi_1)\in{\mathcal S}$ there exists a unique global classical wave map

$$\phi:{\mathbf R}\times{\mathbf R}^2\to{\mathbf H}$$

with $\phi[0]=(\phi_0,\phi_1)$. This is the global regularity problem for energy-critical wave maps from $2+1$-dimensional Minkowski space into the hyperbolic target; the paper and its predecessors aim to prove it, so the conjecture remains open in the supplied source context.