Bopp–Hoff refined balancedness conjecture for relative canonical resolutions

Let $C\subset\mathbb{P}^{g-1}$ be a general canonical curve of genus $g$, let $k$ be a positive integer with Brill–Noether number $\rho(g,k,1)\geq0$, and let $g^1_k$ be a general pencil in $W^1_k(C)$. Write the syzygy bundles as $N_i=\bigoplus_j\mathscr{O}_{\mathbb{P}^1}(a_j^{(i)})$ for $i=2,\dots,\left\lceil\frac{k-3}{2}\right\rceil$. Bopp–Hoff's refined balancedness conjecture. For every such $i$,

$$\max_{j,l}\left|a_j^{(i)}-a_l^{(i)}\right|\leq\min\{g-k-1,i+1\}.$$

Moreover, for every pair of integers $k\geq3$ and $2\leq i\leq\left\lceil\frac{k-3}{2}\right\rceil$, there exists an integer $g$ for which equality holds for the general canonical curve and a general pencil in $W^1_k(C)$. In particular, when $g-k=2$, the relative canonical resolution is balanced. The conjecture concerns the possible imbalance of higher syzygy bundles and is accompanied by the stated sharpness assertion.