There is no universal $F_{\sigma }$ equivalence relation

Rosendal proved that any $K_{\sigma }$ equivalence relation is Borel reducible to ${\mathrm{\ell }}_{\mathrm{\infty }}$. We prove that for every $F_{\sigma }$ equivalence relation $F$ on a Polish space, there is an $F_{\sigma }$ real linear subspace $H$ of ${\mathrm{\ell }}_2$ such that the coset relation $xy$ if and only if $x-y\in H$ has no Baire-measurable reduction to $F$. In particular, there is no $F_{\sigma }$ equivalence relation universal under Borel reducibility for all $F_{\sigma }$ equivalence relations. This answers a question of Rosendal.