Debarre's lower-bound conjecture for Seshadri constants of indecomposable polarized abelian varieties

Let $(A,L)$ be a $g$-dimensional indecomposable polarized abelian variety, meaning that it is not a product of polarized abelian varieties. Let $(J_C,\Theta_C)$ denote the Jacobian of a smooth hyperelliptic curve $C$ of genus $g$, and let $\epsilon(L)$ be the Seshadri constant of $L$. Debarre's conjecture. If $(A,L)$ is not the Jacobian $(J_C,\Theta_C)$ of a smooth hyperelliptic curve $C$ of genus $g$, then

$$\epsilon(L)\geqslant 2.$$

Nakamaye's result gives the general lower bound $\epsilon(L)\geqslant 1$, with equality exactly for a product of an abelian subvariety and an elliptic curve. In higher dimensions, the proposed lower bound remains an open question; the excluded hyperelliptic Jacobians are the known exceptional case motivating the formulation.