Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has objects given by the objects of $C$ and morphisms from $x$ to $z$ given by spans $x \overset{\simeq} \twoheadleftarrow y \twoheadrightarrow z$ where the first arrow is an acyclic fibration and the second arrow is a fibration. The class of weak equivalences $W$ is given by the spans consisting of two acyclic fibrations. The composition of two spans is defined by pullback.

In case it is not obvious, the purpose of the fibration requirement is to ensure that the composite of two such spans is still a span with the backwards arrow an acyclic fibration.

It is well known that we may take a Hammock localization of a relative category in order to obtain a simplicially enriched category. The process involves considering commutative diagrams of zig-zags. Is it the case that the Hammock localization of $(C,W)$ is equivalent to the Hammock localization of $(C_{\operatorname{span}},W_{\operatorname{span}})$?