Koszulness conjecture for the residual tetragonal canonical algebra

Let $C$ be a non-hyperelliptic curve with a base-point free tetragonal system $T$, presented as a complete intersection on a three-dimensional rational normal scroll as in the preceding proposition, with integers $a,b$ satisfying $a+b=g-5$. Assume $K_C(-T)$ is projectively normal, and let $R_{K_C(-T)}$ denote its homogeneous coordinate algebra. Koszulness conjecture. If $a,b\geq 2$, then

$$R_{K_C(-T)}$$

is Koszul. The claim is motivated by the fact that $R_{K_C(-T)}$ is quadratic exactly when $a,b\geq2$; the source notes that it is known when $a=2$ or $b=2$, and explains that this proves the cases $g=9,10$, while the general case remains open.