Decoupling conjecture for smooth surfaces of vanishing Gaussian curvature

Let $S$ be a compact piece of a smooth surface in $\mathbb{R}^n$, and let $N_\delta(M)$ denote its $\delta$-neighborhood. A family of parallelograms has width at least $\delta$ if each parallelogram in the family has width at least $\delta$. Decoupling conjecture. For every $2\le p\le \frac{2(n+1)}{n-1}$ and $0<\delta\ll_S 1$, $N_\delta(M)$ can be $\ell^p(L^p)$ decoupled into parallelograms with width at least $\delta$. Moreover, if $S$ is convex, then this $\ell^p(L^p)$ decoupling can be upgraded to $\ell^2(L^p)$ decoupling. This conjecture would extend the paper's partial decoupling results to arbitrary smooth surfaces in the stated range of exponents, with convexity yielding the stronger decoupling estimate; its resolution is not established in the supplied text.