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We study a class of models for neutrino mass matrix in Type-II seesaw with$A_4$ family symmetry. The resulting neutrino mass matrix can be naturally madeto respect a $\mu-\tau$ exchange plus CP conjugate symmetry (GLS) with the CPviolating phase $\delta$ and the mixing angle $\theta_{23}$ predicted to be$\pm\pi/2$ and $\pi/4$, respectively. When GLS is explicitly broken by complexYukawa couplings, the model predictions for $\delta$ and $\theta_{23}$ can besignificantly modified. Should future experiments will indeed determine$\theta_{23}$ and $\delta_{CP}$ away from the GLS limit values, one then had toconsider models with broken GLS. We study several simple scenarios to show howthe modifications arise when GLS is broken and how future experiments can testthis class of models.

CP violation in neutrino mixing with $\delta = -\pi/2$ in $A_4$ Type-II seesaw model