The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a categorical analogue called braided monoidal category. More precisely, there is an operad in groupoids $\mathcal{P}aB$ whose algebras in the category of categories are braided monoidal category and which is such that a levelwise application of the nerve to $\mathcal{P}aB$ yields an operad equivalent to $\mathcal{D}_2$.

The operad $\mathcal{D}_2$ has a variant called the framed little $2$-disks operad and usually denoted $f\mathcal{D}_2$. The operad $f\mathcal{D}_2$ also has the property that each of its spaces are $K(\pi,1)$'s.

**My question** : Is there a known framed analogue of a braided monoidal category ? More precisely, is there an operad $\mathcal{X}$ in groupoids whose algebras in the category of categories have been defined somewhere in the literature and with the property that levelwise application of the nerve to $\mathcal{X}$ yields an operad equivalent to $f\mathcal{D}_2$.

Note that the question is not about the existence of $\mathcal{X}$ (which is straightforward) but really about the existence of an $\mathcal{X}$ whose algebras have been defined somewhere.