Voisin's gonality conjecture for very general abelian varieties

Let $k\in\mathbb{Z}_{>0}$, and let $A$ be a very general abelian variety. Its gonality is the least degree of a nonconstant map from a smooth projective curve to $A$. Voisin's gonality conjecture. If $A$ has dimension at least $2k-1$, then its gonality is at least $k+1$.

This is Voisin's proposed linear bound on the gonality of very general abelian varieties, improving the known exponential upper bound for the dimension needed to exclude positive-dimensional orbits of degree $k$.