Regular directed complexes are freely generating

Let $P$ be a directed complex. It is freely generating when $(\mathcal{M}o\ell_{}{P})^*$ admits the structure of a polygraph whose $n$-dimensional generators are the $n$-dimensional atoms of $P$. A directed complex is regular if it has the regularity property defined in the paper. Every regular directed complex is freely generating. This conjecture is analogous to, and implies, the corresponding conjecture of Hadzihasanovic; if true, it yields the stated comparison between the associated strict omega-category and the free omega-category on the complex.