Euclidean local smoothing conjecture for wave equations

Suppose $M=\mathbb R^d$ has the flat metric, $u$ solves the Cauchy problem

$$(\partial_t^2-\Delta_g)u(x,t)=0,\qquad u(0,x)=u_0(x),\qquad \partial_tu(x,0)=u_1(x),$$

and define $s_p=\frac{d-1}{2}-\frac{d-1}{p}$ and $p_d=\frac{2d}{d-1}$. Euclidean local smoothing conjecture. For every $p\ge p_d$ and every $\sigma<\frac1p$, there is a constant $C_\sigma$ such that

$$\|u\|_{L^p(\mathbb R^d\times[1,2])}\le C_\sigma\left(\|u_0\|_{L^p_{s_p-\sigma}(\mathbb R^d)}+\|u_1\|_{L^p_{s_p-1-\sigma}(\mathbb R^d)}\right).$$

Here the parameter $s_p$ is the fixed-time regularity order, so the assertion is a gain of $\sigma<1/p$ derivatives from integration in time. The conjecture was verified in dimension $d=2$ by Guth, Wang, and Zhang, while it implies major open problems in higher dimensions.