The hammock-space conjecture for simplicial model categories

Let $\mathcal{M}$ be a simplicial model category and let $A\in\mathcal{M}$ be both fibrant and cofibrant. Let $\widehat{A}_{\bullet}$ be the simplicial set constructed from $A$ in Section~, and let $\widehat{A}$ be its geometric realization.

The hammock-space conjecture. The space $\widehat{A}$ is weakly equivalent to $\emph{Bhaut} A$:

$$\widehat{A}\simeq\emph{Bhaut} A.$$

This conjecture identifies the classifying space constructed from the derived mapping data of $A$ with the classifying space of its simplicial monoid of self-weak equivalences. Its resolution would underpin the subsequent classification of homogeneous functors by homotopy classes of maps into $\emph{Bhaut} A$.