centered moments of a random variable being larger than another

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Suppose that random variables $X,Y$ have equal means, and that for $k=2,3,\ldots$, the $k$-th centered moment of $X$ is no smaller than that of $Y$. Can we say anything about the relation between $X,Y$? For example, does it hold that $X$ is a mean-preserving spread of $Y$?