Gonality–genus bound for the Seshadri constant of a polarized abelian variety

Let $(A,\theta)$ be a polarized abelian variety, and let $C\subset A$ be a smooth curve of genus $h$. Write $c$ for the gonality of $C$ and

$$d=(\theta\cdot C)$$

for its $\theta$-degree. Gonality–genus Seshadri conjecture. Then

$$\varepsilon(\theta)\leq\frac{cd}{c+h-1}.$$

This is proposed as a generalization of the corresponding Jacobian-case bound and is intended to support universal genus estimates for curves on polarized abelian varieties; its general validity is open.