Conjectured sharp orthonormal Strichartz estimate for the wave equation

Let $d \geq 2$, $q \in (2,\infty)$, and $r \in [2,\infty)$ satisfy

$$\frac{1}{q} \leq \frac{d-1}{2}\left(\frac{1}{2}-\frac{1}{r}\right).$$

Let $\beta_{d/2}(q,r)$ denote the exponent appearing in the conjectured estimate, and let the estimate

$$be the orthonormal-data Strichartz estimate for the wave equation. **Conjectured sharp estimate.** The estimate$$

holds in each of the following cases: (i) $\beta \leq \beta_{d/2}(q,r)$ when $\frac{d}{r}>\frac{d-1}{q}$; and (ii) $\beta<\frac{q}{2}$ when $\frac{d}{r}\leq\frac{d-1}{q}$. This conjecture is intended to close the gap between the known sufficient condition and the necessary condition for the orthonormal Strichartz estimate, and would make the sufficient range sharp.