# Haar-$\mathcal{I}$ set and Polish groups

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 $K$?