Square-cap decoupling conjecture for the cone

Let ${\mathcal C}^2=\{(\xi_1,\xi_2,\sqrt{\xi_1^2+\xi_2^2}):\frac14\leq\xi_1^2+\xi_2^2\leq4\}$, and let ${\mathcal N}_\delta({\mathcal C}^2)$ denote its $\delta$-neighborhood. Suppose ${\mathcal P}_\delta({\mathcal C}^2)$ is a partition of this neighborhood into roughly $\delta^{-1}$ near-rectangular boxes $\tau$ of dimensions $\sim\delta^{1/2}\times\delta^{1/2}\times\delta$. For a function $f$ whose Fourier transform is supported in ${\mathcal N}_\delta({\mathcal C}^2)$, write $f_\tau$ for its Fourier restriction to $\tau$. Square-cap decoupling conjecture. For $2\leq p\leq4$,

$$\|f\|_{L^p({\mathbb R}^3)}\lesssim_{\epsilon}(\delta^{-1})^{\frac12-\frac1p+\epsilon}\left(\sum_{\tau\in{\mathcal P}_\delta({\mathcal C}^2)}\|f_\tau\|_{L^p({\mathbb R}^3)}^p\right)^{1/p}.$$

This asks whether the cone admits decoupling into square-like caps, an open question highlighted as one of the most interesting unresolved problems for surfaces in ${\mathbb R}^3$.