"Unwieldy" in this context means that we have to have more precise information about the configurations in the unavoidable set.
The analysis of reducibility in the spline context seems to
require that the topology of the reducible subtriangulation
be specified completely, or in any case in more detail than
in the Four Color Map context.
Daniel P. Sanders mentioned that the e57 edge formation can be omitted from the above unavoidable set.