Restricted-type orthonormal Strichartz estimate on the segment (O,A]

Let $n\geq2$, let $(1/p,1/q)$ lie on the line segment $(O,A]$, and set $s$ by

$$2s=n-\left(\frac{2}{q}+\frac{n}{p}\right).$$

For an orthonormal family $\{f_j\}_{j\in J}\subset\dot{H}^{s}(\mathbb{R}^n)$ and coefficients $\{n_j\}_{j\in J}$, Restricted-type segment conjecture. The estimate

$$\bigg\|\sum_{j\in J} n_j|e^{it\Delta}f_j|^2\bigg\|_{L^q(\mathbb{R}, L^p(\mathbb{R}^n))}\lesssim \|\{n_j\}_{j\in J}\|_{\ell^{q,1}}$$

holds for all such orthonormal families and all coefficient sequences in $\ell^{q,1}$. This is proposed as the analogue, for $(O,A]$, of the endpoint conjecture at $A$ in the segment $[A,D]$; the supplied text gives no resolution.