Bilinear space-time estimate conjecture

Let $N>0$, let $e_1$ be the first coordinate vector, and suppose that the Fourier transforms $\widehat{f_1}$ and $\widehat{f_2}$ are supported on subsets of $\mathbb{B}^n(Ne_1,1)$ such that

$$\operatorname{dist}(\operatorname{supp}\widehat{f_1},\operatorname{supp}\widehat{f_2})\geq\frac{1}{2}.$$

Bilinear space-time estimate conjecture. For $2\leq p,r<\infty$, the estimate

$$\left\||e^{it\Delta}f_1e^{it\Delta}f_2|^{1/2}\right\|_{L_x^pL_t^r(\mathbb{R}^{n+1})}\leq C_{p,r}N^{\frac{1}{p}-\frac{1}{r}}\|f_1\|_{L^2}^{1/2}\|f_2\|_{L^2}^{1/2}$$

holds if and only if

$$\frac{n+2}{p}+\frac{1}{r}\leq\frac{n+1}{2}.$$

This conjecture is formulated because it implies the paper's preceding space-time conjecture. The source does not report a resolution, so the estimate remains open in the stated range.