Small cap decoupling conjecture for the cone

Let ${{\mathbb C}}o^2=\\{(\xi_1,\xi_2,\sqrt{\xi_1^2+\xi_2^2}):1\leq \xi_1^2+\xi_2^2\leq 2\\}$ be the truncated cone, and let $\Gamma(\delta)$ partition its $\delta$-neighborhood into almost rectangular boxes $\gamma$ of dimensions $\sim(\delta^{1/2},\delta,\delta^{1/2})$. Let $\mathcal P_\gamma F$ denote the Fourier projection to $\gamma$. Small cap decoupling conjecture for the cone. If $F:{\mathbb R}^3\to{\mathbb C}$ has Fourier transform supported inside

$${\mathcal N}_{{{\mathbb C}}o^2}(R^{-1}),$$

then for each $2\leq p\leq4$ one has

$$\\|F\\|_{L^p({\mathbb R}^3)}\lesssim_\epsilon R^{\frac12-\frac1p+\epsilon}\left(\sum_{\gamma\in \Gamma(R^{-1})}\\|{\mathcal P}_\gamma F\\|^p_{L^p({\mathbb R}^3)}\right)^{\frac1p}.$$

This conjecture concerns decoupling for the truncated cone, whose Gaussian curvature vanishes identically; the source recalls it as a conjecture stated at the end of the cited work, and no resolution is given here.