Let us take a relative category $(\mathcal{C},\mathcal{W})$, and consider its hammock localization $L_H \mathcal{C}$. It seems to me that for every two objects $X,Y \in \mathcal{C}$ the mapping simplicial set $L_H \mathcal{C}(X,Y)$ has the (strict) right lifting property against all inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ for $n \geq 3$. The reason is the following (if this appears not to be clear enough, I'll edit the question): a diagram of shape $\partial \Delta^n$ means a collection of $n+1$ hammocks of width $n-1$ such that some selected $(n-2)$-faces of these correspond. This compatibility conditions allows us to just take the hammock having all the $n+1$ involved rows, and the vertical arrows will also be uniquely determined by the same compatibility, in that for $n \geq 3$ all required compositions are verified.

There should be a mistake in the above argument, in that we know that there are relative categories with arbitrary mapping simplicial sets, but I failed to find it. Could someone please point it out?