Optimal Strichartz range for compact Lie groups

Let $M$ be a compact Lie group of dimension $d$ and rank $r\geq 2$. Consider the scale-invariant estimate

$$\|e^{it\Delta}f\|_{L^p(I\times M)}\leq C\|f\|_{H^{d/2-(d+2)/p}(M)}.$$

The optimal Strichartz-range conjecture. The estimate should hold on any compact Lie group for every $p>2+\frac{4}{d}$. The paper provides evidence for this assertion for class functions, conditional on the mixed-norm torus conjecture.