Restricted-type endpoint conjecture for orthonormal Schrödinger initial data

Let $d\geq1$, and set

$$(p,q)=\left(\frac{d+1}{d},\frac{d+1}{d-1}\right).$$

For an orthonormal system $(f_j)_j$ in $L^2(\mathbb{R}^d)$ and a sequence $\lambda=(\lambda_j)_j$ in the Lorentz sequence space $\ell^{p,1}(\mathbb{C})$, consider the orthonormal Strichartz expression

$$\sum_j \lambda_j\lvert e^{it\Delta}f_j\rvert^2.$$

Restricted-type endpoint conjecture. The estimate

$$\left\|\sum_j \lambda_j\lvert e^{it\Delta}f_j\rvert^2\right\|_{L^p_tL^q_x}\lesssim\|\lambda\|_{\ell^{p,1}}$$

holds for all such orthonormal systems and sequences.

This endpoint estimate is the remaining critical case in the range $\frac{d+1}{d-1}<q<\frac{d}{d-2}$ discussed for $d\geq3$, where estimates with exponent $\alpha<p$ are known and estimates fail for $\alpha>p$. The conjecture was raised in the cited work and would imply the critical estimate by interpolation.