The restriction type conjecture for the operator $E_R$

Let $n\geq 3$, let $\alpha\in(0,1)\cup(1,\infty)$, and define the restriction-type operator

$$E_Rf(x,t):=\int_{\mathbb{R}^{n-1}}e\left(\frac{R^2}{t}\left|\frac{x-ty}{R}\right|^{\frac{\alpha}{\alpha-1}}\right)\psi\left(\frac{x-ty}{R}\right)f(y)\,dy,$$

where $e(t):=e^{2\pi it}$, $\psi$ is compactly supported and smooth, and $f\in L^1([0,1]^{n-1})$. Restriction type conjecture. For $p>\frac{2n}{n-1}$, $\epsilon>0$, and $R\geq1$, one has

$$\|E_Rf\|_{L^p([0,R]^{n-1}\times[R/2,R])}\leq C_{p,\epsilon}R^{\epsilon}\|f\|_{L^p}.$$

This is a restriction-type formulation related to local smoothing estimates via the pseudo-conformal transformation; the source presents it as a conjectural estimate, and its general validity remains open.