Linear smoothing conjecture for Fourier integral operators

Let $d\geq 2$. Let $T_a^\phi$ be a Fourier integral operator of order $m\in\mathbb{R}$ with a non-degenerate phase function satisfying the cinematic curvature condition. Define

$$\overline{p}_d=\begin{cases}\frac{2(d+1)}{d-1},&d\text{ odd},\\\\[2pt]\frac{2(d+2)}{d},&d\text{ even}. \end{cases}$$

Linear smoothing conjecture for FIOs. For any $p\geq\overline{p}_d$ and $\sigma<\frac{1}{p}$, one has

$$\lVert T_a^\phi f\rVert_{L^p(\mathbb{R}^d\times\mathcal I)}\lesssim \lVert f\rVert_{L^p_s(\mathbb{R}^d)},\qquad s=m+(d-1)\left(\frac12-\frac{1}{p_i}\right)-\sigma.$$

This is the linear local smoothing estimate for Fourier integral operators satisfying cinematic curvature; the supplied text gives no resolution, and the formula contains the undefined index $p_i$, which should be checked against the source.