Is the Hierarchy of Valence Systems with Open Reachability Proper?

Starting from finite state automata, one can create a hierarchy of automata models by repeatedly applying one of the following constructions: Adding VASS counters (i.e. over $\mathbb{N}$, without zero tests), and Building stacks containing the model. As a special case, we get, for example: $\mathsf{FSA}$ with $\mathbb{N}$-counters $= \mathsf{VASS}$ Stacks over $\mathsf{FSA}$ $= \mathsf{PDA}$ $\mathsf{PDA}$ with $\mathbb{N}$-counters $= \mathsf{PVASS}$ It would be interesting to know whether this hierarchy is proper, or if it collapses at some level. A first avenue of investigation might be to consider the hierarchy created by building stacks and adding $\mathbb{Z}$-counters.