  We investigate $\mathbb{C}^\times$-families of flat connections whose leading
term is a nilpotent Higgs field. Examples of such families include real twistor
lines and families arising from the conformal limit. We show that these
families have the same monodromy as families whose leading term is a regular
Higgs bundle and use this to deduce that traces of holonomies are
asymptotically exponential in rational powers of the parameter of the family.
