The initial-object conjecture for pointed update systems on path graphs

Let $\Gamma_n$ be the graph considered in the paper, and let $\mathcal{S}_n^\star$ be the distinguished update system supported on $\Gamma_n$, with preferred state $\star$. A pointed update system is an update system together with a preferred system state. Initial-object conjecture. The pair $(\mathcal{S}_n^\star,\star)$ is an initial object in the category of pointed update systems supported on $\Gamma_n$.

This conjecture gives a categorical approach to determining the universal dynamics of $\Gamma_n$, by identifying a distinguished pointed update system through which pointed update systems supported on the graph should receive a unique morphism. The supplied text does not state a resolution, so the conjecture remains open.