Scattering conjecture for barely supercritical wave equations

Let $p>5$ and set $s_p=\frac{3}{2}-\frac{2}{p-1}$. Consider the defocusing wave equation

$$\partial_t^2u-\Delta u+|u|^{p-1}u=0$$

for initial data $(u_0,u_1)\in\dot H^{s_p}(\mathbb{R}^3)\times\dot H^{s_p-1}(\mathbb{R}^3)$.

Scattering conjecture. If $u$ is the solution with data $(u_0,u_1)$, then $u$ exists for all time $t$ and there is a constant $C_1=C_1\left(\|(u_0,u_1)\|_{\dot H^{s_p}(\mathbb{R}^3)\times\dot H^{s_p-1}(\mathbb{R}^3)}\right)$ such that

$$\|u\|_{L_t^{2(p-1)}L_x^{2(p-1)}(\mathbb{R}\times\mathbb{R}^3)}\leq C_1.$$

This asserts global well-posedness and a global spacetime bound at the scaling-critical regularity for the barely supercritical wave equation; the source presents it as a conjecture because no blow-up is known, while the required global bound and scattering remain unproved.