Naturality conjecture for general paracanonical curves

Let $C$ be a general paracanonical curve of genus $g$ and degree $2g-2$, embedded by a general degree-zero twist of the canonical bundle as

$$C\subseteq\mathbf{P}^{g-2}.$$

Naturality conjecture. The resolution of $C\subseteq\mathbf{P}^{g-2}$ is natural.

This is the weakening of the Prym-Green conjecture obtained by allowing the degree-zero line bundle to be general rather than torsion. The supplied text gives no resolution evidence.