Foam category group completion conjecture

Let $\mathcal{C}_0$ be the decoration data for an $E_1$-monoidal $(\infty,n-1)$-category $\mathscr{F}_{n-1}(\mathcal{C}_0)$ of singular manifolds in $\mathbb{R}^n$, framed in codimension one and decorated by $\mathcal{C}_0$, with singular strata locally modelled on the standard foams $F_{k+1}$. Let $B_n(\mathcal{C}_0)$ be the associated space. Foam group-completion conjecture. The topological group completion of $\mathscr{F}_{n-1}(\mathcal{C}_0)$ is homotopy equivalent to $\Omega B_n(\mathcal{C}_0)$. In particular, for $k<n-1$, $\pi_{k+1}(B_n(\mathcal{C}_0))$ is isomorphic to the group of cobordism classes of closed decorated $k$-foams in $\mathbb{R}^{k+1}$, with cobordisms in $\mathbb{R}^{k+1}\times[0,1]$ and compatible framing and labels; for $i\geq n-1$, the homotopy groups $\pi_i(\Omega B_n(\mathcal{C}_0))$ are represented by the corresponding cobordism classes of $i$-foams, with singular strata locally modelled by $F_{k+1}\times\mathbb{R}^{i-k}$ for some $k<n$. The conjecture is presented as the central motivation of the article, and the authors hope to prove it in subsequent work; its asserted equivalence and the stated homotopy-group descriptions therefore remain open.