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Let $\mathcal{I}$ be a semi-ideal of sets with empty interior on a compact metrizable space $K$. Let an $F_σ$-set $A$ in a Polish group $X$ generically Haar-$\mathcal{I}$.

Then is $A$ always generically Haar-$\mathcal{M}$, where $\mathcal{M}$ is the $σ$-ideal of meager sets in $K$?

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The answer is "yes" and is given in Proposition 12.16 of this paper.

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