On finite non-degenerate braided tensor categories with a Lagrangian subcategory

Let $W$ be a finite dimensional purely odd supervector space over $\mathbb{C}$, and let $\sRep(W)$ be the finite symmetric tensor category of finite dimensional superrepresentations of the finite supergroup $W$. We show that the set of equivalence classes of finite non-degenerate braided tensor categories $\C$ containing $\sRep(W)$ as a Lagrangian subcategory is a torsor over the cyclic group $\mathbb{Z}/16\mathbb{Z}$. In particular, we obtain that there are $8$ non-equivalent such braided tensor categories $\C$ which are integral and $8$ which are non-integral.