In this paper, G. Bujokas and I establish that certain natural vector bundles (specifically the first term in the Casnati-Ekedahl-Schreyer relative canonical resolution) attached to a branched cover of the projective line are usually balanced. The argument proceeds via specialization of the domain curve to a union of rational curves, where the problem then "reduces" to verification of a specific instance of the maximal rank conjecture for certain chains of rational curves.