Planchon's space-time conjecture for the Schrödinger equation

Let $n\geq 1$, let $f\in\mathcal{S}(\mathbb{R}^n)$, and let $e^{it\Delta}f$ denote the solution of the free Schrödinger equation. For $2\leq r,p<\infty$ satisfying

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

Planchon's space-time conjecture. One has

$$\|e^{it\Delta}f\|_{L_x^pL_t^r(\mathbb{R}^{n+1})}\leq C\|f\|_{\dot{H}^{s}(\mathbb{R}^n)},\qquad s=\frac{n}{2}-\frac{n}{p}-\frac{2}{r}.$$

This is the space-time counterpart of the classical Strichartz estimate, with the order of the mixed norms reversed. The paper establishes the sharp global estimate when $n=2$ and improves known critical-case results for $n\geq3$, but the conjecture is not stated as resolved in general.