Equivalence between weak recalages and simplicial triangulations

Let $\mathrm C$ be a sesquicategory. A weak recalage–simplicial triangulation equivalence conjecture. The category of weak recalages of $\operatorname{\bf LN}_+(\mathrm C)$ is equivalent to the category of simplicial triangulations on $\mathrm C$.

This proposes that simplicial triangulations on a sesquicategory are precisely the categorical structures corresponding to weak recalages of its associated category $\operatorname{\bf LN}_+(\mathrm C)$. The source provides examples and related equivalences for complexes, but does not establish the asserted equivalence in general.