Let \(C\subset \mathbb {P}^{g-1}\) be the canonical model of a general curve of genus \(g\). Fix an integer \(k\) such that \(\lceil (g+2)/2\rceil \leq k <g\) and fix a \(g^1_k\) on \(C\). The union of the \((k-1)\)-dimensional linear subspaces spanned by the divisors of the \(g^1_k\) is a rational normal scroll \(\mathbb {P}(\mathcal {E})\). By F. O. Schreyer [Math. Ann. 275, 105–137 (1986; Zbl 0578.14002)] the resolution of \(C\) in \(\mathbb {P}(\mathcal {E})\) is determined by bundles \(N_i\), \(1\leq i\leq k-2\), on \(\mathbb {P}^1\) whose ranks and degrees only depend on \(g\), \(k\), and \(i\). It is conjectured that these bundles have rigid splitting type. This is true if \(k\leq 5\) [A. Deopurkar and A. Patel, Algebra Number Theory 9, No. 2, 459–492 (2015; Zbl 1325.14044)] and [C. Bopp, “Syzygies of 5-gonal canonical curves”, preprint, arxiv:1404.7851]. The \(N_i\) are rigid if \(g = nk+1\) with \(n\geq 1\) and so is \(N_1\) if \(g\geq (k-1)(k-3)\) [G. Bujokas and A. Patel, “Invariants of a general branched cover over the projective line”, Preprint, arxiv:1504.03756]. In the paper under review, the authors prove that for a general \((C,g^1_k)\) with non-negative Brill-Noether number \(N_1\) is unbalaced if and only if \((k-\rho -\frac{7}{2})^2-2k-\frac{23}{4} >0\) and \(\rho >0\), where \(\rho := 2k-g-2\). The authors have many experimental computations (with a dedicated package in Macaulay2) and they make two conjectures. One that the bundle \(N_1\) is balanced for a general \((C,g^1_k)\), when \(\rho (g,k,1)\leq 0\) The other one is a sharp upper bound for the difference between the degree of the maximal rank \(1\) factor and the minimal rank \(1\) factor for a general curve \(C\) and a general \(g^1_k\in W^1_k(C)\).