$L^2$ weak decoupling conjecture for the sphere and paraboloid

Let $P^{n-1}=\{(\xi_1,\ldots,\xi_{n-1},\xi_1^2+\cdots+\xi_{n-1}^2):|\xi_i|\le 1/2\}$, let $S$ be either $P^{n-1}$ or $S^{n-1}$, and let ${\mathcal N}_\delta$ be the $\delta$-neighborhood of $P^{n-1}$ with ${\mathcal P}_\delta$ the corresponding finitely overlapping cover by curved regions; for the sphere, use the analogous decomposition. If $\operatorname{supp}(\widehat f)\subset\mathcal N_\delta$ and $f_\theta$ denotes the Fourier restriction of $f$ to $\theta$, then $L^2$ weak decoupling conjecture. For $p\ge\frac{2(n+1)}{n-1}$ and $\epsilon>0$,

$$\|f\|_p\lesssim_\epsilon \delta^{-\frac{n-1}{4}+\frac{n+1}{2p}-\epsilon}\left(\sum_{\theta\in\mathcal P_\delta}\|f_\theta\|_p^2\right)^{1/2},$$

while for $2\le p\le\frac{2(n+1)}{n-1}$,

$$\|f\|_p\lesssim_\epsilon \delta^{-\epsilon}\left(\sum_{\theta\in\mathcal P_\delta}\|f_\theta\|_p^2\right)^{1/2}.$$

The estimate would imply both the supercritical and subcritical discrete restriction estimates at the critical index; the source notes that the subcritical loss is known to be removable in some low-dimensional cases but regards the general question as extremely difficult.