Ayala–Francis–Rozenblyum's topological stratified homotopy hypothesis

Let $\mathcal{S}\mathrm{trat}$ denote a homotopy theory of topological stratified spaces, and let $\iCat$ denote the homotopy theory of $\infty$-categories. The exit path construction assigns to each topological stratified space an $\infty$-category, with morphisms represented by paths that remain in a stratum or immediately exit to a higher stratum. Ayala–Francis–Rozenblyum's topological stratified homotopy hypothesis. Topological exit paths define a fully faithful functor

$$\operatorname{Exit}\colon \mathcal{S}\mathrm{trat}\hookrightarrow \iCat.$$

This conjecture proposes that the homotopy theory of topological stratified spaces embeds fully faithfully into the homotopy theory of $\infty$-categories via exit paths. Its status is not resolved in the supplied source context.