Rational growth conjecture for torus bundle groups

Let $A\in SL(2,\mathbb{Z})$ and let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$ be the corresponding torus bundle group. Rational growth means that the spherical growth series of $G$ with respect to a finite generating set is a rational function.

Rational growth conjecture. Every torus bundle group $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$ has rational growth with respect to some finite generating set.

The paper proves this for the matrices $A=\begin{bmatrix}0&-1\\1&2k+1\end{bmatrix}$ with $2k+1\geq 5$, extending an earlier result for even traces. The conjecture asks whether the argument applies to all torus bundle groups.