The IFG field curve-index existence conjecture

Let an IFG field mean an infinite, finitely generated field. For a smooth projective geometrically integral curve $C_{/K}$, let $I(C)$ denote its index, and let $g$ be a nonnegative integer and $I$ a positive integer.

IFG field curve-index conjecture. If $K$ is an IFG field and $I\mid 2g-2$, then there exists a curve $C_{/K}$ of genus $g$ with $I(C)=I$.

The divisibility condition is necessary because the canonical divisor has degree $2g-2$, but it is not sufficient over arbitrary fields. The conjecture proposes sufficiency for infinite finitely generated fields; its resolution is not supplied here.