Critical Strichartz estimates for orthonormal systems

Orthonormal Strichartz estimates for the free Schrödinger propagator take the form \begin{equation*} \left\Vert\sum_jλ_j\left|e^{itΔ}f_j\right|^2\right\Vert_{L_t^\frac{q}{2}L_x^\frac{r}{2}(\mathbb{R}\times\mathbb{R}^d)}\lesssim\Vertλ\Vert_{\ell^α(\mathbb{C})} \end{equation*}for arbitrary orthono