Conjecture on diameters of irreducible spherical actions of large cohomogeneity

Let $G$ act irreducibly on the sphere $S^n$ by cohomogeneity $k$, where $n\in 2\mathbb{Z}$. For every $\epsilon>0$, and for all sufficiently large $k$ with $n>k$, the quotient diameter $\operatorname{diam}(S^n/G)$ is within $\epsilon$ of $\pi/2$.

Large-cohomogeneity diameter conjecture. For every $\epsilon>0$ and all sufficiently large $k$ with $n>k$,

$$\left|\operatorname{diam}(S^n/G)-\frac{\pi}{2}\right|<\epsilon.$$

The conjecture concerns the asymptotic behavior of orbit-space diameters for irreducible actions on even-dimensional spheres. The preceding theorem proves exact minimum diameters for specified cohomogeneities and classes of actions, but does not establish this large-cohomogeneity limit.