The universal dynamics monoid conjecture for finite directed acyclic graphs

Let $\Gamma$ be a finite directed acyclic graph. Its universal dynamics monoid $D(\Gamma)$ is the smallest quotient of $\operatorname{\mathbf{HK}}_\Gamma$ through which all evaluation maps from update systems supported on $\Gamma$ factor. Universal dynamics monoid conjecture.

$$D(\Gamma) \simeq \operatorname{\mathbf{HK}}_\Gamma$$

The conjecture extends the proved equality $D(\Gamma_n) \simeq \operatorname{\mathbf{HK}}_{\Gamma_n}=\mathrm{K}_n$ from the family of graphs $\Gamma_n$ to every finite directed acyclic graph. It had been computationally checked for all instances with at most four vertices and for most graphs on five vertices; a general conceptual proof remains open.