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. In fact the pure braid groups themselves are organized into an operad in groups which gives something equivalent to $\mathcal{D}_2$ when applying the classifying space construction. We can see this operad in groups as an operad in categories in the obvious way and consider algebras over that operad in the category of categories. The resulting objects are braided monoidal categories with the special property that the monoidal product is strictly associative. In fact if one wants to encode the most general form of braided monoidal category, there is an operad in groupoids $\mathcal{P}aB$ that does just that.

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.