Minimal-energy blowup solution conjecture for wave maps

Let $\dot{\mathcal H}^1$ be the wave-map energy space, and let an energy-class solution be a restriction of a maximal Cauchy development to a subinterval. Such a solution $\phi:I\to\dot{\mathcal H}^1$ is almost periodic if there exist functions $N:I\to(0,+\infty)$ and $x:I\to\mathbf R^2$, and a compact set $K\subset\dot{\mathcal H}^1$, such that

$${\operatorname{Dil}}_{N(t)}{\operatorname{Trans}}_{-x(t)}\phi[t]\in K$$

for every $t\in I$. Minimal-energy blowup solution conjecture. If the global well-posedness conjecture fails, then there exists an almost periodic maximal Cauchy development $\phi:I\to\dot{\mathcal H}^1$ with non-zero energy. Earlier work established that all almost periodic maximal Cauchy developments have zero energy, so this reduction would rule out the failure of global well-posedness; the existence claim itself is the remaining assertion at this stage.