Sharp Sobolev regularity conjecture for restricted X-ray transforms

Let $d\ge3$ and $1< p < 2$. Suppose $\gamma$ is a smooth nondegenerate curve, and let $\mathfrak R$ denote the associated restricted X-ray transform. With

$$\alpha(p)= \begin{cases} 1-\frac 1p, & 1\le p<\frac{2d}{2d-1},\\ \frac 1{2d}, & \frac{2d}{2d-1} \le p\le 2, \end{cases}$$

Sharp Sobolev regularity conjecture. The operator $\mathfrak R$ boundedly maps $L^p$ to $L_\alpha^p$ for $\alpha \le \alpha(p)$. The sharp $L^p$ regularity estimate is open in higher dimensions for $1<p<2$; in dimension $d=2$, the endpoint remains open for $\frac43\le p<2$.