Internal categories in simplicial sets Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?
Note that this category is closely related to other known models for $(\infty,1)$-categories.  For instance, any simplicially enriched category can be regarded as an internal simplicial category with a discrete simplicial set of objects.  And any internal simplicial category has a bisimplicial nerve which is a Segal space.  One might hope that these functors would be part of Quillen equivalences.
 A: A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence.  In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers.
The above category objects generate a full sub-(relative category) of Rezk's relative category of complete Segal spaces.  As I explain below, Barwick and Kan's work proves that the inclusion of this sub-(relative category) induces an equivalence of homotopy theories.
Barwick and Kan construct a nerve functor $N$ from small relative categories to simplicial spaces. The key point is that anything in the image of this nerve is a category object in the above sense.  
Their nerve functor $N$ has a left adjoint $K$, but they also consider a second functor $M$ from simplicial spaces to relative categories.  The functors $M$ and $N$ are inverse equivalences of homotopy theories in the sense that there is a zigzag of natural weak equivalences
$$NMX \rightarrow NKX \leftarrow X$$ for any simplicial space $X$,
and a natural weak equivalence
$$MNY \rightarrow Y$$
for any relative category $Y$.
If one restricts the domains of $K$ and $M$ to consist only of category objects, the above natural weak equivalences remain intact.  Thus the Barwick+Kan homotopy theory of relative categories is equivalent to the theory of category objects in simplicial spaces.
A: It appears that Geoffroy Horel has solved this problem completely:

Geoffroy Horel, A model structure on internal categories in simplicial sets, Theory and Applications of Categories 30 No. 20 (2015) pp. 704–750 (journal page, arXiv:1403.6873)

