Normal-closure-equivalence of RAAG- and counter-automata

A RAAG (right-angled Artin group) or graph group is defined by a finite undirected graph $(V,E)$ as follows: The resulting group has a generator $x_v$ for every vertex $v \in V$. Whenever two vertices are connected by an edge, then the respective generators are defined to commute. In other words, the group has the presentation $\langle x_v \mid x_vx_w=x_wx_v, (v,w) \in E\rangle.$ By $G$-VASS I mean a finite automaton with an additional "counter" with values in $G$. On each transition the automaton can add a value to the counter, but the counter cannot be accessed; only when the automaton reaches an accepting state and the counter has value $0$ (trivial element of $G$), the input word is accepted. The normal closure of a language $L$ is its closure under concatenation and cyclic rotation. Two languages are nc-equivalent if they have the same normal closure. I know: If the graph of $G$ is not a transitive forest (equivalently: contains a cycle of length $4$ or a line of length at least $4$ as induced subgraph), then the class of $G$-VASS languages is the class of recursive languages. Furthermore, for any $F_2 \times \mathbb{Z}^d$-VASS $\mathcal{A}$ there exists a $\mathbb{Z}^n$-VASS $\mathcal{A}'$ for some $n$ such that $\mathcal{L}(\mathcal{A})$ and $\mathcal{L}(\mathcal{A}')$ are nc-equivalent. Question: Does this extend to $G$-VASS for $G$ being defined by a transitive forest, i.e. is every such $G$-VASS language nc-equivalent to a $\mathbb{Z}^n$-VASS language? A language is $k$-context-free if it is the intersection of $k$ context-free languages. A language is poly-context-free if it is $k$-context-free for some $k$. Question: What is the relation between poly-context-free languages and $\mathbb{Z}^n$-VASS or $G$-VASS languages for RAAGs $G$? What is their nc-equivalence status?