Local smoothing conjecture for curve averages

Let $n \geq 2$ and $2 < p < \infty$. If $\gamma:I \to \mathbb R^n$ is a non-degenerate curve, then the inequality

$$\left( \int_1^2\| A_t f \|_{L^p_\sigma(\mathbb R^n)}^p \, \mathrm{d} t \right)^{1/p} \lesssim_{p,\gamma,\chi} \| f \|_{L^p(\mathbb R^n)}$$

holds for all $\sigma < \sigma(p,n)$, where

$$\sigma(p,n)=\min \left\{ \frac{1}{n}, \frac{1}{n}\left(\frac{1}{2} + \frac{2}{p} \right), \frac{2}{p} \right\}.$$

Local smoothing conjecture. The displayed inequality holds for all $\sigma < \sigma(p,n)$. This conjecture is solved affirmatively for $n=2$; for $n \geq 3$, it is known when $p>4n-2$, while the range $2<p\leq 4n-2$ remains open.