The Street–Roberts conjecture for complicial spaces

Let $(\bDelta^n)$ denote the oriented simplex, let $\tau_{n-1}(\bDelta^n)$ denote its $(n-1)$-truncation, and let $(\Delta^n)^t$ be the stratified simplicial space obtained from the standard simplex with the indicated stratification. A map of stratified simplicial spaces is local with respect to complicial spaces when it induces an equivalence on mapping spaces into every complicial space. Street–Roberts conjecture. For every natural number $n$, the map

$$(\Delta^n)^t \to {\mathrm N}(\tau_{n-1}(\bDelta^n))$$

of stratified simplicial spaces is local with respect to complicial spaces. If true, this would provide a direct comparison between complicial spaces and Segal presheaves on the orientals, and hence facilitate comparisons between models of $\infty$-category theory. The untruncated analogue is stated as a theorem in the paper, but the truncated, marked version remains unresolved in the supplied source.