The presentation conjecture for the monoid of labelled cubes

Let $\overline{\square}([n],[n])$ be the monoid of endomorphisms of the object $[n]$ in the labelled-cube category, and let the generators and relations be those specified in Proposition

$$. **Presentation conjecture.** Proposition$$

gives a presentation by generators and relations of the monoid $\overline{\square}([n],[n])$ for every $n\geqslant 2$.

This conjecture asserts that the stated generators and relations completely describe the endomorphism monoid in every dimension at least two. The supplied text does not establish its resolution status.