Separating weak and full universality for F-sigma relations

Construct an $F_\sigma$ relation with the weak universality property but not the universality property, and determine whether such a relation can induce a $\sigma$-ideal that is provably $\mathbf{\Delta}^1_2$ on $\mathbf\Sigma^1_1$.

Background

The paper gives a $G_{\delta\sigma}$ relation whose induced forcing has the weak universality property but whose relation lacks the full universality property. It then asks whether the analogous separation can occur already for $F_\sigma$ relations, including under the additional definability requirement that the induced ideal be provably $\mathbf{\Delta}^1_2$ on analytic sets.