Bochner–Riesz multiplier conjecture for curved hypersurfaces

Let $\psi:\mathbb R^{n-1}\to\mathbb R$ be smooth with nonvanishing Hessian determinant, let $m^\delta(\xi)=(\xi_n-\psi(\xi'))_+^\delta\chi(\xi')$ for $\delta\geq0$, and let $m^\delta(D)$ denote the corresponding Fourier multiplier. Define

$$\delta(p)=\max\left\{n\left|\frac12-\frac1p\right|-\frac12,0\right\}.$$

Multiplier conjecture. For $1\leq p\leq\infty$, the estimate

$$\|m^\delta(D)f\|_{L^p(\mathbb R^n)}\lesssim_\delta\|f\|_{L^p(\mathbb R^n)}$$

holds whenever $\delta>\delta(p)$. This is the conjectured sharp multiplier range for hypersurfaces with nonvanishing Gaussian curvature; the source does not state a resolution status.