Scaling-critical Strichartz estimates on compact-type symmetric spaces

Let $G/K$ be a Riemannian globally symmetric space of compact type, equipped with the negative of the Cartan--Killing form as the canonical Riemannian metric $g$. Let $d$ and $r$ be the dimension and rank of $G/K$, respectively, and let $I\subset\mathbb{R}$ be a finite time interval. Compact-type symmetric-space Strichartz conjecture. The scaling-critical estimates

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

hold for all $p\geq2+\frac{8}{r}$. The conjecture seeks to extend the character-analysis methods for compact Lie groups to spherical functions on compact-type symmetric spaces; the asserted estimates remain open in this generality.