Exit-path realization conjecture for quasicategories with invertible endomorphisms

Let $K$ be a quasicategory in which all endomorphisms are equivalences. Write $||K||$ for its stratified realization, and let $C_*K$ denote the chain complex associated to $K$. Exit-path realization conjecture. The $\b5$-category of constructible sheaves on the stratified space $||K||$ is categorically equivalent to

$$\mathscr D^{co}(C_*K).$$

This conjecture predicts that the constructible-sheaf category on the stratified realization can be recovered from the chains on $K$. It is presented as being in line with Ayala's conjecture that precisely these quasicategories arise as exit paths of stratified spaces; the source does not provide a resolution.