  We redefine the perfect obstruction theory for $\mathbf{SU}(r)$-Higgs sheaves
constructed by Tanaka-Thomas using a different method in the rank $2$ case: The
key step here is a modification of the $\mathbf{C}^\times$-localisation formula
from Graber-Pandharipande, replacing the torus action by an involution
$(E,\phi) \mapsto (E^*,-\phi^*)$
