Debarre's conjecture on small Seshadri constants of principally polarized abelian varieties

Let $(A,\Theta)$ be a principally polarized abelian variety of dimension $g\geqslant 4$. Here $\epsilon(\Theta)$ denotes the Seshadri constant of the polarization $\Theta$.

Debarre's conjecture. If

$$\epsilon(\Theta)<2,$$

then either $(A,\Theta)$ is decomposable or $(A,\Theta)$ is the Jacobian of a hyperelliptic curve.

This conjecture describes the expected structure of principally polarized abelian varieties with small Seshadri constant. The source states that it remains open; a later work generalizes it to arbitrary polarizations and studies it in small dimensions.