Naively Haar null sets in Polish groups

Let (G,⋅)(G,⋅) be a Polish group. We say that a set X⊂GX⊂G is Haar null   if there exists a universally measurable set U⊃XU⊃X and a Borel probability measure μ   such that for every g,h∈Gg,h∈G we have μ(gUh)=0μ(gUh)=0. We call a set X naively Haar null if there exists a Borel probability measure μ   such that for every g,h∈Gg,h∈G we have μ(gXh)=0μ(gXh)=0. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.