Scaling-critical Strichartz estimates for rational metrics on compact Lie groups

Let $G$ be a compact Lie group equipped with a rational metric $g$. Let $d$ be the dimension of $G$ and $r$ the rank of $G$. Let $I\subset\mathbb{R}$ be a finite time interval. Compact-Lie-group Strichartz conjecture. The scaling-critical estimates

$$\|e^{it\Delta_g}f\|_{L^p(I\times G)}\lesssim\|f\|_{H^{\frac{d}{2}-\frac{d+2}{p}}(G)}$$

hold for all $p>2+\frac{4}{r}$. This is motivated by the corresponding results on tori and the role of rational metrics; the conjecture concerns the expected extension of those estimates to compact Lie groups.