In Definition 4.1.1 of [$(\infty,2)$-Categories and the Goodwillie Calculus I](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), Lurie defines a *weak $\infty$-bicategory* to be a scaled simplicial set that has the extension property with respect to every scaled anodyne morphism. In Theorem 4.2.7, he defines a model structure on $\operatorname{Set}\_{\Delta}^{\operatorname{sc}}$, the category of scaled simplical sets, and in Definition 4.2.8 he defines an *$\infty$-bicategory* to be a scaled simplicial set that is a fibrant object in the model category $\operatorname{Set}\_{\Delta}^{\operatorname{sc}}$.


Every $\infty$-bicategory is a weak $\infty$-bicategory, because every scaled anodyne morphism is a bicategorical equivalence (Proposition 3.1.13). What about the converse? Is every weak $\infty$-bicategory an $\infty$-bicategory?


EDIT:


I think I now have a proof that, indeed, every weak $\infty$-bicategory is an $\infty$-bicategory. I will post it tomorrow.



Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory?
A few years later, it has been shown by [Gagna, Harpaz, and Lanari](https://arxiv.org/abs/1911.01905) that the answer is *yes*. Every weak $\infty$-bicategory is an $\infty$-bicategory.


