The low-regularity local well-posedness conjecture for power nonlinear wave equations

Let $n\geq 4$, let $0\leq s\leq \frac12$, and consider the power nonlinear wave equation

$$\Box u=u^p.$$

The initial data lie in $H^s\times H^{s-1}$. Local well-posedness conjecture. The equation $\Box u=u^p$ is locally well-posed in $H^s\times H^{s-1}$ whenever

$$p\left(\frac{n+1}{4}-s\right)\leq \frac{n+5}{4}-s.$$

This claim concerns the range where the $X^{s,b}$ framework and bilinear estimates are used to improve on basic Strichartz bounds in dimensions $n\geq4$.