Type I finite sample smeariness for random variables on spheres

Let $X$ be a random variable supported on a set $A \subset \mathbb{S}^m$ whose convex closure has nonzero volume and which has a unique mean $\mu$. A point $\mu$ is Type I finite sample smeary when the asymptotic covariance of the sample Fréchet mean exceeds the covariance of the logarithmic map, equivalently when the trace of the former exceeds the variance of $X$. Type I finite sample smeariness conjecture. Under these assumptions, $\mu$ is Type I finite sample smeary. The claim concerns finite-sample smeariness of Fréchet means for distributions on spheres; the supplied text derives it from a Hessian comparison for rotation-invariant distributions, but does not provide a general proof for every random variable satisfying the stated support and uniqueness conditions.