Simplicity conjecture for the perverse sheaves of a smooth theta divisor

Let $(X,\Theta)$ be a principally polarized abelian variety over $\mathbb C$ of dimension $g\geq 2$ with smooth symmetric theta divisor $\Theta$. For the variations $\mathcal V_\pm$ arising from the eigenspaces of the middle cohomology of $Y_x=\Theta_x\cap\Theta_{-x}$, let $\delta_\pm$ be their underlying perverse sheaves on $X$. Simplicity conjecture for the perverse sheaves. If $\Theta$ is smooth, the perverse sheaves $\delta_\pm$ are simple. This is presented as a stronger statement that would imply the simplicity of the variations $\mathcal V_\pm$; the supplied text gives no evidence that it has been resolved.