Determine whether the torus bundle example admits stably exotic fillings

Determine whether the nonorientable 3-manifold constructed as a $T^2$-bundle over $S^1$ with orientation-reversing monodromy represented on $H^1(T^2;\mathbb Z/2)$ by the matrix $\begin{pmatrix}1&1\\1&0\end{pmatrix}$ admits stably exotic fillings for some normal 1-type data.

Background

The paper constructs a nonorientable $T^2$-bundle over $S^1$ whose monodromy has no nonzero fixed points on $H^1(T^2;\mathbb Z/2)$. Consequently, every tangential Pin$^+$ structure induces the same non-bounding spin structure on the fibre, so the manifold does not Pin$^+$-bound and the sufficient criterion used earlier cannot establish stably exotic fillings. The authors leave open whether stably exotic fillings might nevertheless exist using a different normal 1-type.