Global Fréchet mean conjecture for the smeary means in the curse of dimensionality theorem

Let $X$ be a random variable on $\mathbb{S}^m$ whose local Fréchet mean is smeary, as in the curse of dimensionality theorem, and let the corresponding local Fréchet means be those described by that theorem. Global Fréchet mean conjecture. The local Fréchet means constructed in the curse of dimensionality theorem are also global Fréchet means.

Numerical calculations support this claim, but no proof or resolution is given here.