González-Alonso–Torelli maximal-rank conjecture for curves on projective surfaces

Let $S$ be a projective surface, let $L$ be a sufficiently ample line bundle in the sense of Green, and let $C$ be a smooth curve in $|L|$. Here $g$ is the genus of $C$, $q(S)$ is the irregularity of $S$, and the deformation of $C$ within $S$ is the deformation induced by its motion in the linear system.

González-Alonso–Torelli conjecture. Such a deformation can have rank

$$g-q(S).$$

This conjecture predicts that the upper bound for the rank of deformations of curves moving in a sufficiently ample linear system on a surface is attained. The source notes that the assertion is known for smooth curves in $\mathbb{P}^2$ and for $\mathbb{P}^1\times\mathbb{P}^1$, while the general case remains open.