The weak null condition conjecture for quasilinear wave equations

Let $u$ solve a quasilinear wave system of the form

$$\partial_t^2 u-\Delta u=Q(u,\partial u,\partial^2 u),$$

with the weak null condition (WNC) defined by the existence of a global asymptotic-PDE solution for the ansatz

$$u(t,x)\approx\frac{\varepsilon}{|x|}U(q,s,\omega),\qquad q=t-|x|,\qquad s=\varepsilon\log t,\qquad \omega=x/|x|,$$

whose derivatives grow at most exponentially in $s$. Weak null condition conjecture. (WNC) is a sufficient condition for the global regularity of the Cauchy problem with small and localized data. This conjecture concerns whether the asymptotic control encoded by the weak null condition suffices for small-data global existence for general quasilinear hyperbolic systems; the source notes that this sufficiency was unclear.