Representing spaces of $\infty$-stacks

Question

In The stable moduli space of Riemannsurfaces: Mumford’s conjecture, Madsen and Weiss introduce the representing space $|\mathcal{F}|$ of a sheaf of sets $\mathcal{F}$ on the site $\mathscr{X}$ of smooth manifolds as the geometric realization of the simplicial set $[n]\mapsto \mathcal{F}(\Delta^n)$ and prove that concordance classes of elements in $\mathcal{F}(X)$ bijectively correspond to homotopy classes of maps from $X$ to $|\mathcal{F}|$. As also the classifying space of a topological group $G$ admits a description as geometric realization (but now of a simplicial topological space rather than a simplicial set) and concordance classes of principal $G$-bundles on are the same thing as isomorphism classes, this suggests that more generally one should have a representing space for not only sheaves, but also for stacks and higher stacks on $\mathscr{X}$, similarly defined by a topological realization construction, and representing concordance classes. Unfortunately, I have not been able to precisely locate such a statement in the literature, so I'm asking for a reference here (in case such a result has some hope to be correct).

Answer 1

As requested, the comment as an answer:

The case of sheaves valued in ∞-groupoids is the content of a recent preprint of (Berwick-Evans)-[Boavida de Brito]-Pavlov: arxiv.org/pdf/1912.10544.pdf.