About the mathematical instantons with $c_2=4$

In his thesis (1996) Frederic Han gave a description of a dense open set of mathematical instantons for $c_2=4,5$. He assumed that the second order jumping lines are in codimension $3$, but there was a gap in the proof he gave of this fact. I would like to discuss the state of affairs around this in the case $c_2=4$.