The conjecture that all compact spaces feature smeariness

Let $X$ be a compact metric space, and consider random variables taking values in $X$ whose Fréchet means and sample Fréchet means are defined. A distribution is smeary when its Fréchet mean has a limiting convergence rate slower than the classical $n^{-1/2}$ rate. Compact-space smeariness conjecture. All compact spaces feature smeariness: every compact space admits a random variable featuring smeariness. This is stated in the paper as a conjecture and is presented as an open direction; the paper establishes directional smeariness under curvature bounds but does not resolve the assertion for all compact spaces.