I am wondering there is (or is expected to be) an $\infty$-categorical description of the category $Mfld_n$ of $n$-manifolds with morphisms given by the space of codimension zero open embeddings.

For example, here is one possible result in this direction. There is the fully faithful Yoneda embedding $Mfld_n \hookrightarrow Psh(Mfld_n)$, presheaves of spaces on $Mfld_n$. I also found the following result in https://arxiv.org/pdf/1409.0501 that the category of sheaves on $Sh(Mfld_n)$ is equivalent to $Space_{BO(n)}$, spaces with a map to $BO(n)$ (this equivalence takes a space $E\rightarrow BO(n)$ to the sheaf $\phi_E$ with $\phi_E(M)$ equivalent to the space of sections of the pullback $E \times_\tau M$, where $\tau: M \rightarrow BO(n)$ is the tangent bundle classifier). Here, $Sh(Mfld_n)$ is the localization of $PSh(Mfld_n)$ respect to covering morphisms. That is, in $Sh(Mfld_n)$, we add isomorphisms between a smooth $n$-manifold, viewed as a colimit of open disks, and a formal colimit of open disks in $Psh(Mfld_n)$. So if we are allowed to chop up manifolds ``freely", then this category is equivalent to $Space_{BO(n)}$.

Also, https://arxiv.org/pdf/1206.5522 proves that for topological manifolds, factorization homology gives an equivalence between $Fun^{Ex}(Mfld_n, C)$, symmetric monoidal excisive functors from $Mfld_n$ to a symmetric monoidal category $C$, and $E_n-Alg^C$, $E_n$-algebras valued in $C$. Excisive again means we are allowed to chop up our manifolds ``freely."

Question: do we expect (or not) a $\infty$-categorical description of the category $Mfld_n$, without allowing the ``free" chopping up referred to above?

Ideally, such a description would involve some local data, like an $E_n$-algebra, but perhaps that would lead to choppping up $n$-manifolds. If there is no description involving local data, is there description of $Mfld_n$ in terms of something more algebraic?