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Iosif Pinelis
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If, for $n>2$, $\mu_n$ puts masses $n^{-1},n^{-1}, 1-2n^{-1}$ toat $2n, -n, -\frac{n}{n+2}$$2n, -n, -\frac{n}{n-2}$ respectively (so that $\mu$ puts the unit mass toat (-1)$-1$), doesn't it work?

If $\mu_n$ puts masses $n^{-1},n^{-1}, 1-2n^{-1}$ to $2n, -n, -\frac{n}{n+2}$ respectively (so that $\mu$ puts the unit mass to (-1)), doesn't it work?

If, for $n>2$, $\mu_n$ puts masses $n^{-1},n^{-1}, 1-2n^{-1}$ at $2n, -n, -\frac{n}{n-2}$ respectively (so that $\mu$ puts the unit mass at $-1$), doesn't it work?

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Serguei Popov
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If $\mu_n$ puts masses $n^{-1},n^{-1}, 1-2n^{-1}$ to $2n, -n, -\frac{n}{n+2}$ respectively (so that $\mu$ puts the unit mass to (-1)), doesn't it work?