The nef-cone conjecture for the moduli space of principally polarized abelian varieties

Let $g\geq2$. Let ${\cal A}_g^*$ be the Voronoi compactification of the moduli space of principally polarized abelian varieties, let $L$ denote the line bundle of modular forms of weight one, and let $D$ denote the boundary divisor. A divisor on ${\cal A}_g^*$ has the form $aL-bD$.

Nef-cone conjecture. For any $g\geq2$, the nef cone on ${\cal A}_g^*$ is given by the divisors $aL-bD$ satisfying

$$b\geq0\quad\text{and}\quad a-12b\geq0.$$

The statement is known for $g=2$ and $g=3$, where it follows from the theorem given earlier in the paper. The conjecture seeks the same precise description in every genus, and the paper indicates that its theta-function method may extend in principle to higher $g$.