  We present numerical experiments that test the predictions of a conjecture of
Gaiotto-Moore-Neitzke and Gaiotto concerning the monodromy map for opers, the
nonabelian Hodge correspondence, and the restriction of Hitchin's hyperk\"ahler
metric to the Hitchin section. These experiments are conducted in the setting
of polynomial holomorphic differentials on the complex plane, where the
predictions take the form of conjectural formulas for the Stokes data and the
Hitchin metric tensor. Overall, the results of our experiments support the
conjecture.
