Eltzner's Type I finite sample smeariness conjecture 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 Fréchet mean $\mu$. A point $\mu$ is Type I finite sample smeary when the empirical Fréchet mean has a nonstandard finite-sample fluctuation rate at $\mu$ of Type I. Eltzner's conjecture. If $A$ has nonzero-volume convex closure and $\mu$ is the unique Fréchet mean of $X$, then $\mu$ is Type I finite sample smeary. This conjecture predicts finite-sample smeariness under a broad geometric support condition on spheres; the supplied text identifies it as a special case of a conjecture stated in earlier work, but gives no resolution.