Manifold-diagram representation of morphisms

Let $C$ be a presented associative $n$-category. For each $k$, the paper considers the set $Hmor(C)_k$ of $k$-morphisms and the set of $k$-manifold diagrams in the scope of $C$, modulo deformation equivalence.

Manifold-diagram representation conjecture. Let $C$ be a presented associative $n$-category. The mapping from labelled singular cubes to colored cubes descends to an injection of sets

$$Hmor(C)_k \longrightarrow \frac{\operatorname{Set}\{\text{$k$-manifold diagrams in the scope $C$}\}}{\text{deformation equivalence}}.$$

This would identify morphisms with a special class of geometric manifold diagrams, up to deformation equivalence. The source presents it as conjectural and gives no resolution.