One can construct the $d$-dimensional bordism category by declaring the objects to be the $(d-1)$-dimensional compact manifolds without boundary and the morphisms the $d$-dimensional bordisms between them. Call it $\mathcal{Cob}_d$. It is well known that the connected components of the geometric realization of this category are in one-to-one correspondence with $\pi_d(MO)$, where $MO$ is the Thom spectrum for the orthogonal group. This is the classical Thom-Pontryagin theorem.

One can think of constructing a similar category with supermanifolds. Namely, $\mathcal{Cob}_{(d|k)}$ is the category whose objects are $(d-1|k)$-dimensional supermanifolds and the morphisms the $(d|k)$-dimensional bordisms.

Does anyone know of a Thom-Pontryagin type result for this category? Is there a spectrum $MO_{|k}$, whose homotopy groups recover the connected components of the geometric realization of $\mathcal{Cob}_{(d|k)}$?