The discrete localisation conjecture for two-dimensional cobordism categories

Let $\mathcal{N}$, $\mathcal{O}$, $\overline{\mathcal{N}}$, and $\mathcal{K}$ denote the cobordism categories considered in the paper, and let their orientable subcategories and respective localisations be denoted accordingly. Discrete localisation conjecture. On classifying spaces, the canonical maps from $\mathcal{N}$, $\mathcal{O}$, $\overline{\mathcal{N}}$, $\mathcal{K}$, and the orientable subcategories to their respective localisations induce homotopy equivalences. The conjecture asserts that these localisations do not change the homotopy types of the classifying spaces; equivalently, the higher homotopy groups beyond those computed in the preceding results should be trivial. Its status is not established in the supplied text.