Prym-Green conjecture for level paracanonical curves

Let $g\geq 5$, let $[C,\eta]$ be a general point of the moduli space $\mathcal{R}_{g,\ell}$ of pairs consisting of a smooth genus-$g$ curve and an $\ell$-torsion line bundle $\eta\in\operatorname{Pic}^0(C)[\ell]$, with $\ell\geq 2$. The line bundle $\omega_C\otimes\eta$ embeds $C$ as a level $\ell$ paracanonical curve

$$\varphi_{\omega_C\otimes\eta}:C\hookrightarrow\mathbf{P}^{g-2}.$$

Prym-Green conjecture. The resolution of a general level $\ell$ paracanonical curve of genus $g$ is natural.

Here naturality means that in each relevant homological degree at most one of the adjacent graded Betti numbers is nonzero. The conjecture generalizes the syzygetic predictions for Prym-canonical curves and connects resolutions of paracanonical rings with the geometry of Prym varieties. The supplied text gives no resolution evidence.