A counterexample in quasi-category theory

In his pioneering work on the theory of quasi-categories (also known as infinity-categories), Joyal left open the following question: is a morphism of simplicial sets inner anodyne iff it is (i) a monomorphism, (ii) bijective on 0-simplices, and (iii) a weak categorical equivalence? In this talk, I will show that the answer to this question is "no" by giving an example of a morphism of simplicial sets which has these three properties, but which is not inner anodyne. Furthermore, I will use this morphism to refute a plausible description of the fibrations in Joyal's model structure for quasi-categories.