Non-D-finiteness conjecture for Baumslag–Solitar generating functions

Let $\mathrm{BS}(1,M)$ be the Baumslag–Solitar group with $M>1$. Let $G(z;q)$ be the bivariate generating function for the trivial-word enumeration considered in the paper, and let $[q^0]G(z;q)$ denote its constant term in $q$. Non-D-finiteness conjecture. The generating functions $G(z;q)$ and $[q^0]G(z;q)$ are not D-finite. This is motivated by the $q$-deformed Catalan structure of the related generating functions, but no proof is given.