Hyperelliptic characterization by the Seshadri constant

Let $(A,\theta)$ be an indecomposable principally polarized abelian variety of dimension $g\geq 4$. The hyperelliptic characterization conjecture. If $\varepsilon(A,\theta)<2$, then $(A,\theta)\simeq (JC,\theta_C)$ for a hyperelliptic curve $C$. This predicts that, in dimension at least four, an indecomposable principally polarized abelian variety with Seshadri constant below $2$ must be the Jacobian of a hyperelliptic curve.