The conjecture that smeariness begets finite sample smeariness

Let $X$ be a random variable on a metric space. A distribution is smeary when its Fréchet mean has a limiting convergence rate slower than the classical $n^{-1/2}$ rate, and it has finite sample smeariness when finite-sample mean variance exhibits the corresponding excess over the Euclidean benchmark. Smeariness-begets-finite-sample-smeariness conjecture. Whenever there is a random variable featuring smeariness, there are nearby random variables featuring finite sample smeariness. The paper presents this as a conjectural general phenomenon and discusses results establishing it near smeary distributions in particular settings; the general assertion remains open.