Scattering below the ground state for the Zakharov system

Let $(f,g)\in H^1\times L^2$ satisfy

$$4\mathcal{E}_Z(f,g)<\|Q\|_{\dot{H}^1}^2, \qquad \|g\|_{L^2}\leqslant \|Q\|_{\dot{H}^1}.$$

Here $\mathcal{E}_Z$ denotes the Zakharov energy and $Q$ is the ground state. Scattering below the ground state conjecture. There exists a unique global solution $(u,V)\in C(\mathbb{R},H^1\times L^2)$ to the Zakharov system with initial data $(u,V)(0)=(f,g)$ such that

$$\|u\|_{L^2_tW^{\frac{1}{2},4}_x(\mathbb{R}^{1+4})}<\infty.$$

This conjecture would establish scattering for all energy-space data below the ground-state threshold in four dimensions. The surrounding results provide global well-posedness below this threshold and scattering criteria, while the required finiteness of the dispersive Strichartz norm is not known in general; scattering is known in the radial case.