Sogge's local smoothing conjecture for the wave equation

Let $u$ solve the wave equation Cauchy problem

$$(\partial_t^2-\Delta)u(x,t)=0,$$

with $u(x,0)=f(x)$ and $\partial_tu(x,0)=0$ on $\mathbb{R}^d\times\mathbb{R}$. For $s_p=\frac{d-1}{2}-\frac{d-1}{p}$, Sogge's conjecture. For all $p\geq\frac{2d}{d-1}$,

$$\|u\|_{L^p(\mathbb{R}^d\times[1,2])}\lesssim_{\varepsilon}\|f\|_{L^p_{s_p-1/p+\varepsilon}(\mathbb{R}^d)}.$$

This conjecture predicts a gain of $1/p$ derivative from local time averaging. The source presents it as a conjecture, although the supplied parser marks it resolved; the precise resolution and scope should be checked.