Sharp maximal estimate conjecture for orthonormal systems of wave equations

Let $n\geq2$ and $\frac12<s<\frac n2$. Let $(f_j)_j$ be an orthonormal system in $H^s(\mathbb{R}^n)$, and let $\beta$ be the exponent in the maximal estimate denoted by

$$. **Maximal estimate conjecture.** The estimate$$

holds for every such orthonormal system if and only if

$$\beta\leq \min\Big\{\frac n{n-2s},\frac {n-1}{n+1-4s}\Big\}.$$

This conjecture is proposed as partial progress toward the optimal maximal estimates for orthonormal systems in the wave-equation setting. The stated range is motivated by counterexamples from the single-datum problem, while the paper establishes only partial results toward the conjectured sharp bound.