The dimension-seven conjecture for tritangents of space sextics

Let $C$ be a smooth space sextic contained in a singular quadric. Let $\alpha=\theta-\theta_0\in \mathrm{J}(C)[2]\backslash\{0\}$, where $\theta$ is one of the $120$ odd theta characteristics and $\theta_0$ is the vanishing even theta characteristic. The space $W_{C,\alpha}$ is the associated space appearing in the tritangent construction.

Dimension-seven conjecture.

$$\dim W_{C,\alpha}=7.$$

The conjecture is motivated by the paper's computation showing this dimension for an example and by the observation that it occurs for every $\alpha$ arising from one of the $120$ odd theta characteristics in that example. Its general validity for smooth space sextics contained in singular quadrics is not established in the supplied text.