Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface: The space of trace-free Higgs fields, whose determinant have only simple zeros, is open and dense in the space of all trace-free Higgs fields.

I would like to see a proof, a reference or a counter-example.

Motivation: This property is certainly true for a generic stable bundle, as one might see by investigating the Hitchin fibration.