Bilinear smoothing conjecture for Fourier integral operators

Let $d\geq2$. Let $T_a^{\phi_1,\phi_2}$ be a bilinear Fourier integral operator whose non-degenerate phase functions satisfy the cinematic curvature condition, and let $a\in S^m_{1,0}$ be compactly supported in $(x,t)\in\mathbb{R}^d\times\mathbb{R}$, of order $m<0$. Define

$$\frac1p=\frac1{p_1}+\frac1{p_2},\qquad m=m_1+m_2,$$

with $m_1\leq0$ and $m_2<0$. Bilinear smoothing conjecture for FIOs. For $p_1,p_2\geq\overline p_d$ and $\sigma_i<\frac1{p_i}$, one has

$$\lVert T_a^{\phi_1,\phi_2}(f,g)\rVert_{L^p(\mathbb{R}^d\times\mathbb{R})}\lesssim \lVert f\rVert_{L^{p_1}_{s_1}(\mathbb{R}^d)}\lVert g\rVert_{L^{p_2}_{s_2}(\mathbb{R}^d)},$$

where

$$s_i=m_i+(d-1)\left(\frac12-\frac1{p_i}\right)-\sigma_i.$$

This is proposed as the bilinear analogue of the linear local smoothing conjecture in dimensions $d\geq2$; the supplied text gives no evidence that it has been resolved.