Asymptotic growth conjecture for trivial words in Baumslag–Solitar groups

Let $\mathrm{BS}(N,N)$ be the Baumslag–Solitar group, and let $g_{n,0}$ denote the number of trivial words of length $n$ in it. Write $\mu_N$ for the exponential growth rate and $A_N$ for the corresponding amplitude. Asymptotic growth conjecture. The number of trivial words in $\mathrm{BS}(N,N)$ satisfies

$$g_{n,0} \sim A_N\mu_N^n n^{-2}$$

for $N\geq 2$. The exponent $n^{-2}$ is established in the paper for $N=2,3,4,5$, but the amplitudes have not been identified and the general assertion remains open.