D-finiteness conjecture for Baumslag–Solitar generating functions

Let $\mathrm{BS}(N,M)$ be a Baumslag–Solitar group, and let $G(z;q)$ be the generating function for trivial-word enumeration considered in the paper. Write $[q^0]G(z;q)$ for its constant term in $q$. D-finiteness conjecture. The generating functions $G(z;q)$ and $[q^0]G(z;q)$ are D-finite only when $N=M$. Thus, for $N\neq M$, the paper conjectures that these generating functions are not D-finite. This extends the preceding conjecture for $\mathrm{BS}(1,M)$ and remains open.