Joyal's cylinder conjecture

For simplicial sets $A$ and $B$, let $\mathbf{Cyl}(A,B)$ be the category of cylinders from $A$ to $B$. Thus an object is a simplicial set $X$ equipped with a map $X\to\Delta[1]$ whose fibres over $0$ and $1$ are $A$ and $B$, respectively. The category has terminal object $A\star B$, so every cylinder has a canonical morphism $X\to A\star B$. A morphism in $\mathbf{Cyl}(A,B)$ is between fibrant cylinders when both its source and target are fibrant in Joyal's model structure. Joyal's cylinder conjecture. A cylinder $X\in\mathbf{Cyl}(A,B)$ is fibrant if and only if the canonical morphism

$$X\longrightarrow A\star B$$

is an inner fibration, and a morphism between fibrant cylinders in $\mathbf{Cyl}(A,B)$ is a fibration if and only if it is an inner fibration. This characterizes the fibrant objects and fibrations between them in the Joyal model structure on cylinders; the source describes the statement as an open question, while the abstract reports that the paper proves it.