Krämer–Weissauer's Tannakian Schottky conjecture

Let $\mathcal{J}_g$ denote the locus of $g$-dimensional Jacobians in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties, and let $\mathcal{J}^\mathrm{fake}_g$ be the closure of the locus where the Tannakian group and representation associated to the theta divisor have the form occurring for Jacobians. Krämer–Weissauer's Tannakian Schottky conjecture. For $g\geq 0$, we have

$$\mathcal{J}_g=\mathcal{J}^\mathrm{fake}_g.$$

Krämer's result shows that the Jacobian locus is an irreducible component of the fake-Jacobian locus, giving a weak solution to the Schottky problem; the conjecture asserts that no additional points occur.