High-dimensional $p$ decoupling conjecture for smooth hypersurfaces

Let $\phi:\mathbb{R}^{n-1}\to\mathbb{R}$ be smooth, and let $\mathcal{M}_{\phi}=\{(\xi,\phi(\xi)):\xi\in[0,1]^{n-1}\}$. A set $S\subset[0,1]^{n-1}$ is $(\phi,\delta)$-flat when it satisfies the flatness condition defined in the paper. For a measurable function $f:\mathbb{R}^n\to\mathbb{C}$ and measurable $S$, let $f_S$ denote the Fourier restriction of $f$ to $S\times\mathbb{R}$. High-dimensional $p$ decoupling conjecture. For every $\epsilon>0$, there is a sufficiently large $A=A(\epsilon)$ such that, for every $\delta>0$, one can find a finitely overlapping collection $\mathcal{S}_{\delta}$ of subsets of $[0,1]^{n-1}$ satisfying

$$\sum_{S\in\mathcal{S}_{\delta}}\chi_S\leq C_{\epsilon}\log(\delta^{-1}),$$

with every $S$ $(\phi,A\delta)$-flat, and such that for $2\leq p\leq\frac{2(n+1)}{n-1}$,

$$\|f\|_{L^p}\leq C_{\epsilon}\delta^{-\epsilon}(\#\mathcal{S}_{\delta})^{\frac12-\frac1p}\left(\sum_{S\in\mathcal{S}_{\delta}}\|f_S\|_{L^p}^p\right)^{\frac1p}$$

for all $f$ whose Fourier supports are contained in $N_{\delta}(\mathcal{M}_{\phi})$. This is proposed as a possible extension of $\ell^p$ decoupling to higher dimensions; the statement is conjectural for general smooth $\phi$.