Improved Strichartz range on compact-type symmetric spaces

Let $G/K$ be a Riemannian globally symmetric space of compact type with the canonical metric and let $d$ and $r$ denote its dimension and rank. Let $I\subset\mathbb{R}$ be a finite time interval, and define

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

Improved symmetric-space Strichartz conjecture. Under these assumptions, this estimate holds for all $p>2+\frac{4}{r}$. This strengthens the preceding conjectured range $p\geq2+\frac{8}{r}$ and is presented as an analogue of the compact-Lie-group conjecture; it remains open.