Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ordered set of its cells, where the ordering is given by inclusions. I have been wondering about the claim that the $\infty$-category of exit-paths $\operatorname{Sing}^S (X)$ is equivalent to (the nerve of) $S$, but I couldn't find a reference for this - is it true?

As for some more background and indications on why this should hold: First of all, for conically stratified spaces, there always is a conservative functor from the exit-path category into the stratification poset $\operatorname{Sing}^S(X) \to N(S)$. The fibers of this map are, by "Higher Algebra" A.7.5, given by the homotopy types of the strata, which in out case are contractible (they are just open balls and points respectively).

The paper "Constructible hypersheaves via exit paths" claims in the beginning of 4.1 that for a locally finite simplicial complex $X'$ with poset of simplices $S'$, the map $\operatorname{Sing}^{S'} (X') \to N(S')$ is an equivalence. It also references HA A.7.5 for this claim, however I don't really understand how this shows the claim, in particular why this map is fully faithful - could someone explain this?

Finally, this paper in 4.2 also gives references that show that a regular CW complex is conically stratified, and proves that for a locally countable regular CW complex, exodromy holds: $$ \operatorname{Sh}^{cbl} (X) \simeq \operatorname{Fun}(\operatorname{Sing}^S (X), \mathcal{S})$$ It however does not explicitly describe the exit-path category, which could however be done using their methods if the finite case is understood.

Given a finite, regular CW complex $X$ (by regular, I mean that the gluing maps $D^n \to X$ from the closed unit ball to $X$ are homeomorphisms onto their image), denote by $S$ the finite partially ordered set of its cells, where the ordering is given by inclusions. I have been wondering about the claim that the $\infty$-category of exit-paths $\operatorname{Sing}^S (X)$ is equivalent to (the nerve of) $S$, but I couldn't find a reference for this - is it true?

As for some more background and indications on why this should hold: First of all, for conically stratified spaces, there always is a conservative functor from the exit-path category into the stratification poset $\operatorname{Sing}^S(X) \to N(S)$. The fibers of this map are, by "Higher Algebra" A.7.5, given by the homotopy types of the strata, which in out case are contractible (they are just open balls and points respectively). In fact, A.6.10 even tells us that for a simplicial simplicial complex stratified by its poset of simplices, the canonical map from the exit-path category into the nerve of the stratification poset is an equivalence. My hope is that the requirement on a CW complex to be regular is similar to the restriction from arbitrary simplicial sets to simplicial complexes (or maybe even semi-simplicial sets).

Finally, the paper "Constructible hypersheaves via exit paths" in 4.2 gives references that show that a regular CW complex is conically stratified, and proves that for a locally countable regular CW complex, exodromy holds: $$ \operatorname{Sh}^{cbl} (X) \simeq \operatorname{Fun}(\operatorname{Sing}^S (X), \mathcal{S})$$ It however does not explicitly describe the exit-path category, which could however be done using their methods if the finite case is understood.