Surjectivity conjecture for products of sections of odd theta characteristics

Let $\mathcal{C}$ be the curve under consideration, let $\kappa$ be the theta characteristic used in the construction, and fix $v\in \operatorname{Pic}(\mathcal{C})[2]$. For every effective odd theta characteristic divisor $D$ such that $D+v$ is also an effective odd theta characteristic divisor, multiplication of sections gives a map

$$\bigoplus_{D,\,D+v}\mathcal{L}(D)\otimes \mathcal{L}(D+v)\longrightarrow \mathcal{L}(\kappa+v).$$

Surjectivity conjecture. The map is surjective when $D$ and $D+v$ range over all effective odd theta characteristic divisors. The conjecture would ensure that the conditional step in the algorithm can be controlled computationally by the resulting linear-dependence test; its status is not established in the supplied text.