Free-generation conjecture for constructible directed complexes

Let $P$ be a directed complex. Say that $P$ is freely generating if $(\mathcal{M}o\ell_{}{P})^*$ admits the structure of a polygraph whose $n$-dimensional generators are the $n$-dimensional atoms of $P$. Free-generation conjecture. Every constructible directed complex is freely generating. The claim is unproved in the source. It is false for general directed complexes, where known counterexamples involve looping phenomena excluded by the constructibility condition.