Recognition of the general type-(4,3) fundamental group as a torus-bundle group

Determine whether the fundamental group presented for the genus-3 Lefschetz pencil $X_\phi$ associated with an arbitrary matrix $\begin{pmatrix}p&q\\r&s\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ can be recognized as the fundamental group of a torus bundle over a torus.

Background

For a general mapping class determined by a matrix in $\mathrm{SL}(2,\mathbb{Z})$, the paper gives a presentation for the fundamental group of the resulting pencil $X_\phi$. In several special cases, the author identifies the group with the fundamental group of a torus bundle and consequently determines the homeomorphism type.

The remark explicitly records that the general presentation has not been recognized as a torus-bundle group. Resolving this question could extend the homeomorphism-type analysis from the computed special cases to the full family of type-$(4,3)$ constructions.