Small-data critical well-posedness conjecture for one-dimensional wave maps

Consider the wave-map Cauchy problem for maps into the sphere $S^{m-1}\subset {\text{\bf R}}^m$, with initial data $f$ satisfying $f(x)\in S^{m-1}$ and zero initial velocity. In the critical case, use the homogeneous Sobolev space $\dot H^{\frac{n}{2}}$, with constant functions assigned zero norm.

Small-data critical well-posedness conjecture. The Cauchy problem is well-posed for small data in $\dot H^{\frac{n}{2}}$.

Well-posedness is known for Sobolev regularity $s>n/2$, while the critical case is described as very difficult and only partially understood. The conjecture concerns the scaling-invariant endpoint regularity.