Syzygy shifting for canonical bundles on symmetric powers

Let $C$ be a smooth complex projective curve of genus $g\geq 2$, let $C_k$ denote its $k$-th symmetric power, and let $K_C$ be the canonical bundle. Say that a line bundle $L$ on $C$ satisfies syzygy shifting to dimension $k$ if, whenever $L$ has property $N_p$, the associated line bundle

$$N_L:=\operatorname{det}(L^{[k]})$$

on $C_k$ has property $N_{p-(k-1)}$. Here property $N_p$ means that $L$ has linear syzygies to order $p\geq 0$. Syzygy-shifting conjecture. The canonical bundle $K_C$ satisfies syzygy shifting to dimension $k$ when

$$\operatorname{Cliff}(C)\geq k.$$

This proposes that the syzygetic behavior of the canonical curve shifts predictably to canonical embeddings of symmetric powers, extending the projective-normality result for $C_2$ and the relationship between Clifford index and syzygies. The supplied text gives no resolution or partial cases, so the conjecture remains open.