Fourier extension conjecture for the truncated paraboloid

Let $P^{n-1} = \{\xi_n = \sum_{j=1}^{n-1} \xi_j^2,\ |\xi_j| \leq 1,\ \forall 1 \leq j \leq n-1\}$ be the truncated unit paraboloid, let $\mathrm{d}\sigma$ be its natural surface measure, and let $\widecheck{f\,\mathrm{d}\sigma}$ denote the inverse Fourier transform of the measure $f\,\mathrm{d}\sigma$. Fourier extension conjecture. For $p > \frac{2n}{n-1}$ and every $f \in L^{\infty}(P^{n-1},\mathrm{d}\sigma)$,

$$\|\widecheck{f\,\mathrm{d}\sigma}\|_{L^p(\mathbb{R}^n)} \lesssim \|f\|_{L^{\infty}}.$$

By duality, this is equivalent to Stein's Fourier restriction conjecture for the same paraboloid and exponent range. It is a central Fourier restriction problem and remains open in general.