Signature-refined local smoothing conjecture for Fourier integral operators

Let $\mathcal{F}$ be a Fourier integral operator on $\mathbb{R}^n\times[1,2]$ with symbol of order $\mu$, satisfying H1) and H2). Suppose that each associated cone has signature $\kappa$, meaning that the absolute difference between the numbers of positive and negative principal curvatures is $\kappa$, where $0\leq\kappa\leq n-1$. Define

$$\bar{p}_{n,\kappa}=\begin{cases}2\,\frac{\kappa+2(n+1)}{\kappa+2(n-1)},&n\text{ odd},\\[8pt]2\,\frac{\kappa+2n+3}{\kappa+2n-1},&n\text{ even}. \end{cases}$$

Signature-refined local smoothing conjecture. There is $1/p-$ local smoothing for $\mathcal{F}$, meaning that the local smoothing estimate holds for every $\sigma<1/p$, for all $\bar{p}_{n,\kappa}\leq p<\infty$. The conjecture refines the general FIO threshold according to the signature of the cone; the supplied text presents it as a conjectural consequence of modified Bourgain examples, with no resolution stated.