Universal decoupling conjecture for real analytic surfaces

Let $S$subset${\mathbb R}^3$ be the graph of a nonconstant real analytic function $g:[-1,1]^2\to{\mathbb R}$. For $0<\delta\leq 1$, let ${\mathcal N}_\delta(S)$ be its $\delta$-neighborhood, and let $f_\tau$ denote the Fourier restriction of $f$ to a box $\tau$ in a partition of this neighborhood. Universal decoupling conjecture. There is a partition ${\mathcal P}_\delta(S)$ of ${\mathcal N}_\delta(S)$ into essentially flat boxes $\tau$, possibly of different dimensions, such that, whenever the Fourier transform of $f$ is supported in ${\mathcal N}_\delta(S)$,

$$\|f\|_{L^4({\mathbb R}^3)}\lesssim_{\epsilon}\delta^{-\epsilon}|{\mathcal P}_\delta(S)|^{\frac14}\left(\sum_{\tau\in{\mathcal P}_\delta(S)}\|f_\tau\|_{L^4({\mathbb R}^3)}^4\right)^{1/4}.$$

This proposes a universal $l^4$ decoupling for arbitrary real analytic surfaces, with the main difficulty being the choice of essentially flat boxes when curvature degenerates.