Bopp–Hoff balancedness conjecture for the first syzygy bundle

Let $\mathscr{H}_{g,k}$ denote the Hurwitz space of degree-$k$ covers $C\to\mathbb{P}^1$, and let $\rho(g,k,1)$ be the Brill–Noether number. For a general cover, let $N_1$ be the first syzygy bundle in its relative canonical resolution. Bopp–Hoff's first-syzygy balancedness conjecture. If $\rho(g,k,1)\leq0$, then $N_1$ is balanced. The authors report verification of this conjecture for $g\leq23$, while the general assertion remains open.