Pareschi–Popa adjoint-module conjecture for indecomposable ppav's

Let $(A,\Theta)\in {\mathscr A}_g({\mathbb C})$ be an indecomposable principally polarized abelian variety that is neither the Jacobian of a curve nor the intermediate Jacobian of a cubic threefold. Let $\mathscr{A}\hspace*{-0.15em}d_\Theta$ denote the clean holonomic ${\mathscr D}$-module whose Tannakian representation is the adjoint representation associated with the theta divisor. A summand is a subobject $\mathscr M\hookrightarrow\mathscr{A}\hspace*{-0.15em}d_\Theta$.

Pareschi–Popa adjoint-module conjecture. Every summand $\mathscr M\hookrightarrow\mathscr{A}\hspace*{-0.15em}d_\Theta$ is either a skyscraper sheaf or has support $A$.

This criterion would imply the conjecture characterizing theta-divisor summands: lower-dimensional non-skyscraper constituents of the adjoint module are expected to occur only for Jacobians of curves and intermediate Jacobians of smooth cubic threefolds. The source presents the statement as an implication of the preceding conjecture, and no resolution is given.