Suppose we are given a category enriched over **semi-simplicial** sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this category all **inner** Kan conditions are satisfied, i.e. we have $\mathrm{Kan}(n,k)$ for all $n$ and $1\le k \le n-1$, but not necessarily $\mathrm{Kan}(n,0)$ and $\mathrm{Kan}(n,n)$.

Is there any "canonical" way to make this category become enriched over **simplicial** sets, i.e. to find "canonical up to equivalence" degeneracy maps, so that the category will become an $(\infty,1)$-category?

There are several papers on making a Kan-simplicial set out of Kan-semi-simplicial one, including the classical paper by Rouke and Sanderson and a more recent by McClure, but I haven't found anything concerning quasi-categories.