Pencil decomposition conjecture for residual tetragonal line bundles

Let $C$ be a non-hyperelliptic curve of genus $g$ with a base-point free tetragonal system $T$, and suppose $C$ has the complete-intersection presentation on a three-dimensional rational normal scroll described above. Let $K_C(-T)$ be the residual line bundle. Pencil decomposition conjecture. Under these assumptions, there exists a decomposition

$$K_C(-T)\cong A+B,$$

where $A$ and $B$ are base-point free pencils of degree $g-3$. This conjecture is proposed as a method for proving the preceding Koszulness conjecture and is therefore stronger than it; the source gives no resolution.