NLKG scattering conjecture below the ground-state threshold

Let $\mu=\pm1$. Let $(u_0,u_1)\in H^1_x\times L^2_x$. In the focusing case, assume additionally that $M(u_0)<M(Q)$ and $E(u_0,u_1)<E(Q)$, where $Q$ is the ground state. A solution scatters if there exist $(u_0^\pm,u_1^\pm)\in H^1_x\times L^2_x$ such that its difference from the corresponding free Klein–Gordon evolution tends to zero in $H^1_x\times L^2_x$ as $t\to\pm\infty$. NLKG conjecture. There exists a global solution $u$ with initial data $(u_0,u_1)$ satisfying

$$\|u\|_{L^\infty_tH^1_x}+\|u_t\|_{L^\infty_tL^2_x}+\|u\|_{L^4_{t,x}}\leq C(E(u_0,u_1)),$$

and consequently $u$ scatters both forward and backward in time. This is the expected global well-posedness and scattering statement for the two-dimensional cubic Klein–Gordon equation below the ground-state threshold; the supplied text gives no resolution status.