Function-field Freiman conjecture for spaces of small combinatorial genus

Let $K$ be an algebraically closed field and let $F$ be an extension field of $K$. Let $S$ be a finite-dimensional $K$-subspace of $F$ such that $K\subset S$. Define the combinatorial genus $\gamma$ of $S$ by

$$\dim S^2=2\dim S-1+\gamma.$$

Assume that $\gamma\leqslant\dim S-3$. Function-field Freiman conjecture. If $g$ is the genus of the field $K(S)$, then

$$g\leqslant\gamma,$$

and there exists a Riemann–Roch space $L(D)$ containing $S$ such that

$$\dim L(D)\leqslant\dim S+\gamma-g.$$

This is a function-field analogue of Freiman-type structure results: spaces with small product growth should be controlled by Riemann–Roch spaces on curves of genus bounded by their combinatorial genus. The source presents the assertion as a conjecture; no resolution is supplied here.